Strong electronic coupling to lattice degrees of freedom in crystals is one of the most important mechanisms by which scientifically interesting and technologically important materials properties arise. Clarifying the interactions between phonons and electrons is critical to understanding the mechanisms responsible for a variety of exotic phenomena in materials, including cooperative and dynamical Jahn-Teller effects [1][2][3][4], giant magnetocapacitance [5,6], and multiferroicity [7], and to developing new functional materials to control future devices [8].
The rare-earth sesquioxides (RESO) are a class of chemically simple materials that are nevertheless excellent laboratories for observing the complex interactions between electronic and lattice degrees of freedom [5,6,9,10]. Because the 4 forbital states of the lanthanide sesquioxides are spatially localized on the ionic nuclei, these states are only weakly split by the surrounding crystal field environment. Weak crystal field splitting leads to low-lying electronic excitations that are close in energy to lattice vibrations, providing an opportunity for significant interaction and/or hybridization between electronic and phonon excitations. In the presence of a sufficiently strong interaction, as in CeAl 2 [11,12], CeCuAl 3 [13], Ce 2 O 3 [6], TbInO 3 [14], or Ho 2 Ti 2 O 7 [15], the degree of mixing between phonon and electronic excitations can lead to new excitations with significant mixed character, known as vibronic excitations.
Many materials having strong electron-phonon coupling exhibit anomalies in their vibrational spectra. For example, the phonon energies in most cubic rare-earth sesquioxides (RESOs) vary linearly with the lattice parameter [23], as shown in Fig. 1. However, the energies of many Ramanactive phonons in Eu 2 O 3 and Yb 2 O 3 deviate from these trends [23][24][25]. Specifically, the phonons having energies between 250 cm -1 and 510 cm -1 in Eu 2 O 3 and Yb 2 O 3 are significantly softer than predicted by the trend exhibited by other RESOs [23][24][25], while the highest-energy phonon in Yb 2 O 3 has a markedly higher energy than would be expected by the trend. Previous reports of the Raman and IR spectra of Eu 2 O 3 and Yb 2 O 3 by Urban and Cornilsen suggest that the anomalous phonon energies in these materials result from weaker chemical bonding and shallower interatomic potentials [25]. By contrast, Schaack and Koningstein propose that low-energy crystal field excitations couple to the phonons, leading to the anomalous phonon behavior [24]. Yet another explanation for these phonon anomalies has been proposed more recently, particularly in the case of Eu 2 O 3 , wherein oxygen vacancy defects are attributed to the soft phonon behaviors [23]. However, Eu 2 O 3 lattice dynamical calculations provide evidence that disputes this hypothesis [26].
Raman scattering studies of rare-earth sesquioxides offer a powerful method for elucidating the origins of CEF-phonon coupling in these materials, generally, and for testing the theories of phonon anomalies in Eu 2 O 3 and Yb 2 O 3 , more specifically. In this paper, we report a study of electronphonon coupling effects in Eu 2 O 3 and Yb 2 O 3 , focusing on CEF-phonon coupling effects, using temperature-dependent (3-300 K) and magnetic-field-dependent (0-7 T) Raman scattering experiments. We can clearly discern the positions of all the strong Raman-active phonons and confirm the energies of many of the low-lying crystal electric field (CEF) electronic excitations in Eu 2 O 3 and Yb 2 O 3 . Our measurements reveal evidence of defect-induced behavior in Eu 2 O 3 , which may explain the anomalous phonon softening in this material. We find evidence that CEF-phonon coupling in these materials is important for understanding anomalous temperature dependences of phonons having energies between 250-650 cm -1 FIG. 1. Summary plot of the energies of the Raman-active phonons for various C-type RESOs as measured by the authors at T = 3 K, as functions of the lattice constant of the RESO. Trend lines are linear fits of the data, omitting Yb 2 O 3 and Eu 2 O 3 , to highlight the anomalous behavior of the two materials. Lattice constant data from Refs. [16][17][18][19][20][21][22].
in both Eu 2 O 3 and Yb 2 O 3 . However, our results suggest that CEF-phonon coupling in Eu 2 O 3 is weaker than in Yb 2 O 3 and largely arises because of lattice vacancy defects which have been reported to occur in Eu 2 O 3 crystals [27][28][29]. While we do not find evidence for defects in the spectra of Yb 2 O 3 , we do observe that Yb 2 O 3 exhibits significant anomalies in phonon energies, intensities, and line widths as functions of temperature and magnetic field, indicative of strong coupling between phonons and low-lying CEF electronic excitations in this material.
Polycrystalline samples of cubic Eu 2 O 3 and Yb 2 O 3 were prepared from commercially available powders (Alfa Aesar) with 99.996% and 99.99% purities, respectively. Disks of pressed powder were sintered in a box furnace under ambient atmosphere at 1000 • C for 36 h before being fractured into small fragments. This high-temperature treatment is necessary to eliminate contamination from rare-earth carbonates, hydrates, and hydroxides formed by reactions between the oxides and carbon dioxide and water in the atmosphere. Sample fragments were stored under positive nitrogen pressure in a desiccator box to minimize contamination. Inelastic light (Raman) scattering measurements were performed using the 647.1 nm excitation of a Coherent Innova 70C Kr + laser. The scattered light was collected in a backscattering geometry by dispersing through a custom triple-grating spectrometer and recorded with a liquid nitrogen-cooled charge-coupled-device detector. Samples were held in a continuous He-flow cryostat and horizontally mounted in the open bore of a superconducting magnet. This experimental arrangement allows Raman scattering measurements to be conducted under simultaneous control of the temperature (3-300 K) and magnetic field (0-7 T). Spectra obtained were in the energy range of approximately 20-955 cm -1 . All magnetic field-dependent measurements were performed in the Faraday geometry with the incident light wave vector parallel to the magnetic field. The incident light was circularly polarized, and scattered light of all polarizations was collected.
The RESOs comprised of rare-earth constituents heavier than Sm crystallize in the cubic, C-type structure with space group Ia 3 [24]. The unit cell of this structure features two types of symmetry inequivalent rare-earth ion sites, occupying the 8b and 24d Wyckoff positions, with C 2 and S 6 symmetries, respectively, and oxygen ions in the 48e sites [24]. The 22 zone-center, Raman-active phonons for this structure are given by Raman = 14A g + 4E g + 4T g . However, previous Raman spectroscopic studies of both Eu 2 O 3 and Yb 2 O 3 have observed only some of the Raman-allowed phonons [23,24,[30][31][32]. Figure 2 obtained at 297 K (top, black curves) and 3 K (bottom, red curves). We observe several phonons in the Yb 2 O 3 and Eu 2 O 3 spectra that have not been previously reported. Because we measured polycrystalline samples, we were unable to perform the polarization-dependent studies necessary to completely determine the symmetries of the phonons. However, previous studies of single-crystal C-type RESOs [24] and lattice dynamical calculations performed by Łażewski et al. [26] allow us to tentatively assign symmetries to all the observed phonons. We present these assignments in Table I, alongside comparisons to previous Raman scattering studies of Eu 2 O 3 and Yb 2 O 3 . Future studies of these materials would greatly benefit from single crystals, allowing more clear determination of the symmetries of the excitations.
In the spectra obtained at 3 K, we observe several additional features that are consistent with electronic transitions between CEF-split electronic states of the rare-earth ions in both Eu 2 O 3 and Yb 2 O 3 . These features are energetically consistent the measured energies of transitions observed in luminescence experiments in Eu 2 O 3 and with the energies of CEF levels obtained from fitting Schottky anomalies in Yb 2 O 3 heat capacity data [33]. Additionally, we are able to rule out that these features are spin-wave excitations, because there is no long-range magnetic order in these materials. Eu 2 O 3 and Yb 2 O 3 exist in their paramagnetic phases over the entire ranges of temperature, pressure, and magnetic field explored in these experiments [34,35]. We are also able to rule out that these are additional phonon modes on the basis that there is no experimental or theoretical evidence for temperature-dependent structural transitions in either Eu 2 O 3 or Yb 2 O 3 between T = 3-300 K. Finally, the dramatic temperature-dependent increase in spectral weight with decreasing temperature of these features is consistent with their electronic origin, owing to increasing population of the initial state (the ground state) associated with the transitions.
The energies of the observed CEF transitions are summarized in Table II, along with the expected symmetries of the excited states in cases where the symmetry of the state can be confidently determined and results from previous experiments and calculations [10,33]. In the following, we present the spectroscopic features observed in (Sec. III A) Eu 2 O 3 and (Sec. III B) Yb 2 O 3 , we summarize the dependences of these spectroscopic features on temperature and applied magnetic field, and we discuss what these results reveal concerning the significant roles of strong CEF-phonon coupling in these materials.
Our results on Eu 2 O 3 presented in Figs. 2(a)-2(c) reveal narrow phonons in the 3 K spectra, consistent with published spectra of other C-type RESOs [23,24], and our own
TABLE I. Energies of Raman-active phonons in Eu 2 O 3 and Yb 2 O 3 in cm -1 and their irreducible representations (IR). The symmetry of each phonon in this table is given by the associated irreducible representation (IR) of the space group Ia 3 in the rightmost column. The second and fifth columns are the results of this study. For results published by other authors, the symmetry analyses of their results have not been altered in this table, despite disagreements between the results of density functional theory (DFT) and Raman studies. Eu 2 O 3 Energy from DFT a Eu 2 O 3 Energy from Raman Eu 2 O 3 Energy from Raman b Eu 2 O 3 Energy from Raman c Yb 2 O 3 Energy from Raman Yb 2 O 3 Energy from Raman d IR a 92.634 95.2 -94 96.1 99.2 T g 97.116 --109 102 -T g 115.535 120.2 -119 120 121 A g 130.337 136.4 -134 134.3 -T g 140.698 146.6 -145 145.5 -E g 168.224 178.148 --175.5 -T g 173.005 182.6 -175 179.2 -T g 277.061 -----T g 285.673 287.3 266.4 289 279.7 279 T g 292.68 299.7 --306.5 305 E g 304.218 -----T g 313.276 319.6 --331.2 -A g 327.793 -----T g 330.202 -----E g 348.414 340.4 336 339 357.6 358 T g 373.941 387.8 -385 415.5 420 A g 381.566 396.3 380 -430.3 443 T g 407.908 423.5 -425 507 -T g 477.2 -459 ---T g 513.872 543 --584 -A g 520.226 560 --602 -E g 535.489 569 -559 618 621 T g a Results from Ref. [26]. b Results from Ref. [30]. c Results from Ref. [23]. d Results from Ref. [24].
TABLE II. Energies of low-energy CEF excitations in Eu 2 O 3 and Yb 2 O 3 in cm -1 . The results in the second and fourth columns are from this study. Eu 2 O 3 Energy from Luminescence a Eu 2 O 3 Energy from Raman Yb 2 O 3 Energy from Raman b Yb 2 O 3 Energy from Raman R 3+ Ion Site Symmetry 139 139 S 6 217 217 C 2 334 336 C 2 351 -C 2 388 376 S 6 423 -S 6 537 -C 2 595 601 C 2 740 -S 6 980 -S 6 1021 -C 2 a Results from Ref. [10]. b Results from Ref. [33].
unpublished results on RESOs. Of the 22 Raman-allowed phonons in Eu 2 O 3 , we observe 17 phonons in Eu 2 O 3 , several more than have been previously reported. The phonon energies we measure agree well with the energies predicted using density functional theory calculations up to an energy of approximately 340 cm -1 [26]. For observed phonons in Eu 2 O 3 having energies higher than 340 cm -1 , the predicted energies underestimate our measured energies by up to 35 cm -1 .
The ground state of a free Eu 3+ ion is unique amongst the RE ions in that it is singlet state, with term symbol 7 F0. Additionally, the first excited level is much closer in energy to the ground state than in many RE ions, being separated by only approximately 300 cm -1 (≈40 meV). The relatively low first-excited state CEF energy holds true for the Eu 3+ ions in Eu 2 O 3 as well, based on our Raman scattering results and earlier luminescence spectroscopy results [10]. Given the site symmetries of the Eu 3+ ions in Eu 2 O 3 , the states and symmetries of the excited 7 F1 manifolds of the two types of symmetry inequivalent RE ions can be determined without knowing the associated energy levels. In C 2 symmetry, the degeneracy of the 7 F1 manifold is fully lifted by the CEF, yielding one state with A symmetry given by |J z = 0 , and two states with B symmetry that are given by linear combinations of the J z eigenstates |J z = +1 ± |J z = -1 . In S 6 symmetry, the degeneracy of the 7 F1 manifold is only partially lifted, resulting in two levels-one with A u symmetry, given by the state |J z = 0 , and the other with E u symmetry, comprised of the degenerate J z eigenstates |J z = +1 and |J z = -1 .
In our Raman spectra of Eu 2 O 3 , we expect to observe three transitions from the ground state to the excited 7 F1 levels of the C 2 Eu 3+ ions, given that all these transitions are Raman-allowed. Transitions from the ground state to the excited levels of the 7 F1 manifold of the S 6 Eu 3+ ions are parity forbidden, given that these states occur with ungerade symmetry, and are therefore Raman forbidden. Unexpectedly, our results presented in Fig. 3 show that we observe two resonances having energies and temperature dependences consistent with transitions between the ground and excited CEF states [10]. We can confidently assign the 217 cm -1 excitation to the CEF electronic transition from the ground state to the first-excited CEF level of the C 2 Eu 3+ ions, based on analyses of the luminescence intensity ratios by Buijs et al. [10]. Further, we tentatively assign the 139 cm The CEF excitations are in the regions shaded in orange. There is a phonon at 136.4 cm -1 , which is nearly coincident with one CEF excitation. Additionally, there is a weak sideband to the 217.3 cm -1 CEF excitation at 214.9 cm -1 with similar temperature dependence, suggesting that it is a related feature. Individual spectra have been offset vertically for clarity. to the Raman-forbidden electronic transition from the ground state to the first-excited CEF level of the S 6 ions (see Fig. 4), based on the same analysis as previously mentioned [10]. There is no evidence in the spectra obtained at 3 K for any higher-energy CEF transitions associated with either the C 2 or S 6 sites. Based on these measurements, we propose a level diagram, shown in Fig. 4, for the CEF splitting of the 7 F1 manifold of the C 2 and S 6 symmetry ions in Eu 2 O 3 . The energies of these CEF excitations and the Eu 3+ ion site they are associated with are summarized in Table II.
The anomalous presence of the Raman-forbidden 139 cm -1 CEF transition suggests either a breakdown of the Raman selection rules in this system, or that the Eu 3+ ions previously reported to occupy the S 6 site instead occupy a lattice site with lower symmetry. Beyond the two clear CEF transitions, we also observe a small secondary feature in our Raman spectra of Eu 2 O 3 , shown in Fig. 3, having an energy of approximately 214 cm -1 at 3 K. The presence of this additional peak suggests that some of the C 2 site Eu 3+ ions experience a different CEF environment, leading to a lower-energy first-excited level. We propose that structural defects prevalent in crystals of Eu 2 O 3 [27][28][29] are the likely origin of the breakdown of Raman selection rules, specifically regarding the 139 cm -1 parity-forbidden transition associated with the S 6 site and the presence of the additional 214 cm -1 transition. We discuss the substantial evidence for the presence of defects through their dramatic impact on the spectra of Eu 2 O 3 in the following section.
Our temperature-dependent Raman scattering measurements of Eu 2 O 3 reveal anomalous softening of the 396.4 and 423.5 cm -1 phonons with decreasing temperature, shown in Fig. 5. Previous studies of Eu 2 O 3 have identified a CEF excitation having an energy of 423 cm -1 [10]. We propose that a weak vibronic coupling exists between the 396.4 and 423.5 cm -1 phonons and this 423 cm -1 CEF excitation, leading to an avoided level repulsion between these coupled excitations. Given that shared symmetry is an essential component of vibronic coupling, and that the 7 F1 states of the S 6 ion have ungerade symmetry rather than the gerade symmetry associated with the Raman-active phonons, vibronic coupling between these modes can occur only if there is a lowering of the symmetry of the S 6 site-specifically, a loss of inversion symmetry facilitated by structural defects.
In addition to the anomalous softening of the 396.4 and 423.5 cm -1 modes, we observe additional features in spectra obtained at temperatures near 300 K, shown in Fig. 6. These features have broad line widths and dramatically temperaturedependent intensities, losing intensity as the temperature is lowered and disappearing in spectra obtained below 200 K. Given this dramatic temperature dependence, we can rule out two-phonon excitations as an explanation of these features.
While the 87 and 134 cm -1 features are nearly energetically coincident with transitions originating from the first-excited levels of the Eu 3+ ions, we can largely rule out that these features are transitions originating from thermally populated excited CEF levels. This explanation for the presence of these peaks fails to account for the 234 cm -1 feature, for which there is not an energetically coincident CEF transition between excited CEF levels. Additionally, if these features were transitions originating from excited states, we would expect to observe the 87 and 234 cm -1 peaks down to a lower temperature than the 134 cm -1 peak.
A more convincing interpretation of these features is suggested by luminescence measurements of Eu 2 O 3 -and Eu 3+ -doped Y 2 O 3 and Gd 2 O 3 , where energy transfer reactions are observed [10]. In materials where the electronic excitations are localized by disorder, energy transfer reactions can occur, whereby the energy associated with an electronic excitation at one lattice site is transferred to another lattice site [10,36]. In resonant cases, this can happen without mediation by phonons. However, if there is an energy mismatch between the excitations associated with the two lattice sites, phonons having the appropriate energy can mediate the transfer of energy from one lattice site to the other [10]. The energies of these strongly temperature-dependent features in our Raman spectra of Eu 2 O 3 (87, 134, and 234 cm -1 ) are well aligned with the energies reported to be involved in energy transfer reactions between 5 D levels in Eu 2 O 3 [10]. The energies of these features also closely match those calculated for three infrared-active phonons, which have the appropriate ungerade symmetries required to facilitate energy transfer from C 2 to S 6 symmetry Eu 3+ ions [26]. Further, the dramatic temperature dependence of these features is more consistent with the temperature dependence of energy transfer reactions observed in Eu 3+ -doped Y 2 O 3 and Gd 2 O 3 , where the reaction rate goes as R ∼ T 7 [10]. Consequently, we argue that in addition to the other defect-induced effects described above, the presence of defects in Eu 2 O 3 facilitate energy-transfer coupling between the 5 DJ electronic manifolds and infrared-active phonons of Eu 2 O 3 leading to a breakdown in the Raman selection rules.
We can confidently identify nineteen of the 22 Ramanactive phonons in Yb 2 O 3 based on our measurements performed at 3 K, presented in Figs. 2(d)-2(g). As previously reported by Schaack and Koningstein [24], the phonons having energies greater than 250 cm -1 in our Yb 2 O 3 spectra are broadened even at 3 K, while those having lower energies display narrow line widths similar to those of the other C-type RESOs. This intrinsic broadening at low temperatures has been previously attributed to strong electron-phonon coupling-arising either from the energetic proximity of CEF levels to these phonons [24] or from modulation of the interatomic potential by shallow electronic levels below the Fermi level [25]. Indeed, we show in the following sections that Yb 2 O 3 exhibits several CEF excitations that are nearly resonant with these phonons, and the energetic proximity of these CEF excitations to the phonons create favorable conditions for vibronic coupling between these degrees of freedom.
A free Yb 3+ ion has an eightfold degeneracy associated with its ground state, given by the term symbol 2 F7/2. However, in a crystal, this degeneracy is lowered by the electrostatic potential produced by the ions surrounding a given Yb 3+ ion, yielding a manifold of CEF levels that are only weakly split in energy. The ground states of both the 8b and 24d Yb 3+ ions in Yb 2 O 3 are split into manifolds of four doubly degenerate levels. These low-lying Yb 2 O 3 CEF states are well approximated by admixtures of J z eigenstates of the Yb 3+ free ion 2 F7/2 manifold. In this approximation, the S 6 states take on one of two generalized forms: the first, shown in Eq. ( 1), has A 3/2g symmetry; the second, shown in Eq. ( 2), has E 1/2g symmetry. On the other hand, the C 2 states take on the form given by Eq. ( 3), which has E 1/2 symmetry. In Eqs. ( 1)-( 3), the kets are labeled by their J z eigenvalues and coefficients A-I are factors determined by diagonalization of the CEF Hamiltonian associated with each site to yield appropriately normalized states:
Our Raman spectra of Yb 2 O 3 reveal excitations having energies of 336, 376, and 601 cm -1 , presented in Figs. 7 and 8, which we assign to CEF electronic excitations from the ground state to excited levels of the 2 F7/2 manifolds of both the C 2 and S 6 ions (see Fig. 9). This assignment is consistent with previous analyses by Schaack and Koningstein [24] and Gruber et al. [33]. These CEF excitations are absent or nearly absent at room temperature and gain substantial intensity as the temperature is decreased, consistent with the expectation that CEF transitions from the ground state to these excited states gain intensity as the ground state CEF level becomes less thermally depopulated at low temperatures. Based on the analysis performed by Gruber et al. [33], we can confidently assign the 336 and 602 cm -1 excitations to electronic transitions from the ground state to the first-and second-excited levels of the C 2 symmetry ions, and we assign the 376 cm -1 excitation to the electronic transition from the ground state to the first-excited level of the S 6 symmetry ion. Based on our measurements, we propose a level diagram for the 2 F7/2 manifold of the C 2 and S 6 symmetry ions in Yb 2 O 3 , shown in Fig. 9. The energies of the CEF excitations measured in this and previous studies, as well as the associated Yb 3+ ion sites, are summarized in Table II.
FIG. 8. Temperature dependence of the features having energies greater than 400 cm -1 , showing the dramatic evolution of phonons in this region with decreasing temperature. The CEF transition at 602 cm -1 is, again, highlighted in orange. Individual spectra have been offset vertically for clarity. FIG. 9. Proposed low-energy CEF level schemes for both Yb 3+ ion sites in Yb 2 O 3 . Arrows depict the observed transitions from the ground state to excited states. Energies presented in parentheses are of levels not observed in this study, but instead are obtained from Ref. [33].
Additional supporting evidence for the electronic character of the 336, 376, and 601 cm -1 excitations can be found in their magnetic-field dependences. As these levels are Kramers doublets-that is, time-reversal symmetry paired states having opposite angular momentum-a magnetic field is expected to cause these levels to split by an amount that depends linearly on the strength of the magnetic field. As shown in Fig. 10 and summarized in Fig. 11, we observe that the splitting of Kramers doublets with applied magnetic field is monotonic and gradual. We also observe that both modes that originate from the 376 cm -1 transition anomalously increase in energy with increasing field. We will later discuss how the anomalous magnetic field dependence of the energy and splitting of the 376 cm -1 CEF transition may result from coupling to phonon degrees of freedom.
Transitions from the ground states to excited 2 F7/2 states of both the C 2 and S 6 symmetry ions are Raman-allowed by group theory. However, contrary to the reports by Gruber et al. [33], we do not observe any features having an energy of approximately 740 cm -1 . Measurements were not performed beyond 955 cm -1 in this study, so we are unable to validate the CEF excitation energies from the ground state to the third CEF levels of either the C 2 (1021 cm -1 ) or S 6 (980 cm -1 ) ions reported by Gruber et al. [33].
Our temperature-dependent studies of the Yb 2 O 3 spectra reveal three distinct types of anomalies: (i) phonons having energies between 400-600 cm -1 with dramatic temperature dependent intensities; (ii) reentrant electronic excitations that gain and then lose intensity with decreasing temperature; and (iii) soft phonons whose energies decrease with decreasing temperature. Previous Raman scattering reports on Yb 2 O 3 have remarked on the weak intensities and broad line widths of phonons having energies greater than 200 cm -1 in spectra obtained at or near room temperature [24,[30][31][32]37]. Our results show that these linewidth anomalies persist over a large temperature range between 3-300 K.
Further, our results presented in Fig. 8 show that above 400 cm -1 , only two clear Raman-active features appear in spectra obtained at room temperature. However, as the temperature is lowered, we observe five additional strong excitations having energies greater than 400 cm -1 that emerge and gain intensity. These include five phonons having energies of 415.5, 430.3, 507, 617 cm -1 , and one CEF electronic transition having an energy of 602 cm -1 , which we discussed in Sec. III B 2. The assignment of the 602 cm -1 mode as a crystal field transition, rather than a phonon mode, is supported by earlier analyses of spectroscopic data by Gruber et al. [33]. Other C-type RESOs feature several Ramanactive phonon modes having energies greater than 400 cm -1 [24,[30][31][32]37], but these phonons do not exhibit dramatic temperature dependences similar to those observed in Yb 2 O 3 .
In contrast to the excitations in Yb 2 O 3 that gain significant intensity with decreasing temperature, the excitation centered at approximately 702 cm -1 loses intensity below 90 K and is ultimately absent in spectra obtained at 30 and 3 K. Several other features gain intensity as the temperature is lowered, before losing intensity as the temperature is reduced further, beyond 90 K, and are ultimately absent in spectra obtained at 3 K. The most apparent of these reentrant excitations have energies of 256.3, 294, 347, 584 cm -1 . Several other excitations having larger line widths and weaker, reentrant intensities are also evident with energies of approximately 481, 520, 541, 563 cm -1 . While the temperature dependences of these features suggest that they may be electronic excitations between excited CEF states of the Yb 3+ 7 F 7/2 manifolds, we can rule out this interpretation because these features do not have energies coincident with any of the transitions between the Yb 3+ 7 F 7/2 levels. We can rule out resonant scattering between FIG. 11. Summary plot of the magnetic field dependence of the energies of magnetically active modes in the Yb 2 O 3 spectra. Only features that split under an applied magnetic field are included in this summary plot.
excited CEF levels, because of the large Yb 2 O 3 band gap, which is significantly greater than the excitation energy of our laser [38]. Further, we can likely rule out energy transfer reactions, like those we attribute to the anomalous features in Eu 2 O 3 , as the reentrant temperature dependences of these features are inconsistent with the expected temperature dependences for energy transfer reactions [10]. Unfortunately, based on our existing Raman measurements, we are unable to offer a definitive explanation for the anomalous reentrant temperature-dependent behavior of these features, suggesting a potential avenue for future study. Polarization-dependent Raman scattering or momentum resolved measurement techniques such as inelastic neutron scattering performed on single crystals may aid in elucidating the origin of these features.
Finally, we observe evidence for strong CEF-phonon coupling in Yb 2 O 3 in both our temperature-and magnetic field-dependent studies. As shown in Fig. 7, four phonons having energies 279.7, 306, 330.8, and 358 cm -1 at 3 K anomalously soften by approximately 6 cm -1 (2.1%), 3 cm -1 (1%), 4.5 cm -1 (1.3%), and 6 cm -1 (1.6%), respectively, from their room-temperature energies as the sample is cooled. Note that heat capacity measurements show no evidence for a structural transition below room temperature under ambient pressure conditions in Yb 2 O 3 [33]. Consequently, the phonon softening we observe in this material is not likely associated with structural phase transition.
Similar phonon softening observed in hexagonal, A-type RESOs, Ce 2 O 3 [6] and Pr 2 O 3 [9] is well explained by the model of vibronic coupling proposed by Thalmeier and Fulde [11]. Thalmeier-Fulde-type coupling between these soft phonons and the nearby CEF excitations in Yb 2 O 3 is consistent with both the temperature-dependent softening and the deviation from the trends in phonon energies established by many of the other RESOs [11]. In the Thalmeier-Fulde picture, near-resonance and shared symmetries between phonon and CEF electronic excitations are necessary criteria for vibronic coupling. Given the energetic proximity of the four soft phonons to the 336 and 376 cm -1 CEF transitions, the resonance criterion is well satisfied for these excitations in Yb 2 O 3 [11]. Further, all Raman-active phonons have overlapping symmetry with transitions between states of the J = 7/2 manifold of the C 2 symmetry Yb 3+ ions. While we are unable to conclusively identify the symmetries of the ground or excited states of the S 6 symmetry Yb 3+ ions, the 358 cm -1 T g phonon fulfills the symmetry criterion for all of the allowed transitions between states of the J = 7/2 manifold of the S 6 symmetry ions. Therefore, we find that the necessary and sufficient conditions for Thalmeier-Fulde-type vibronic coupling are well satisfied in Yb 2 O 3 .
Thalmeier-Fulde-type coupling leads to mixing of the phononic and electronic character of the coupled excitations, renormalizing the bare excitations into hybrid vibronic states. As shown in Fig. 10 and summarized in Fig. 11, we find additional evidence for hybrid electronic-phononic character of the 357, 415.5, 429, and 507 cm -1 modes based on their anomalous magnetic field dependences. Specifically, as the magnetic field is increased, we observe splitting of these phonons and level-repulsion behavior between these phonons and the nearby CEF electronic excitations as they shift to higher energies. Further, the 306.5 cm -1 peak develops an asymmetric line shape at high fields, while the 279.7 cm -1 peak narrows significantly with increasing field.
Splitting of these phonons with external magnetic fields suggests there is coupling between the lattice and some spin-or other angular momentum-degree of freedom in Yb 2 O 3 . Given that Yb 2 O 3 exists in the paramagnetic phase at T = 3 K [34], we can rule out coupling to spin-wave excitations. Additionally, we rule out an interpretation involving coupling to spins of conduction electrons because Yb 2 O 3 is an insulator. Consequently, the observed splitting of these phonons is most consistent with the hybrid electronicphononic character of these modes [39,40], offering additional support that Thalmeier-Fulde-type vibronic coupling significantly impacts the low-energy electronic and phonon spectra of Yb 2 O 3 .
In summary, the results presented here allow us to identify specific mechanisms responsible for anomalous phonon behavior in rare-earth sesquioxides and illustrate the varied manifestations of the coupling between phonons and low-energy electronic excitations in these materials. Upon comparing the spectra of these two systems, we find that different mechanisms are likely responsible for the observed phonon softening in Yb 2 O 3 and Eu 2 O 3 . Specifically, we find evidence of strong Thalmeier-Fulde-type CEF-phonon vibronic coupling in Yb 2 O 3 based on our observations of temperature-dependent phonon softening, field-dependent splitting of phonons, and avoided-level repulsion behavior between electronic excitations and phonons. In contrast, we find evidence of defect-mediated electron-phonon coupling in Eu 2 O 3 , based on the presence of Raman-forbidden, infraredactive phonons and electronic excitations in our measured spectra. Looking forward, we expect that Yb 2 O 3 and Eu 2 O 3 are not unique amongst the RESOs in displaying the hallmarks of strong electron-phonon coupling. Many other RESOs display subtle anomalous phonon behavior, reflecting diverse, novel realizations of interactions between the lattice and electrons associated with the close proximity and shared symmetries of phonons and rare-earth derived CEF electronic excitations prevalent in these materials.