Coupling of lattice, spin and intra-configurational excitations of Eu3+ in Eu2ZnIrO6

In Eu2ZnIrO6, effectively two atoms are active i.e. Ir is magnetically active, which results in complex magnetic ordering within the Ir sublattice at low temperature. On the other hand, although Eu is a van-vleck paramagnet, it is active in the electronic channels involving 4f 6 crystal-field split levels. Phonons, quanta of lattice vibration, involving vibration of atoms in the unit cell, are intimately coupled with both magnetic and electronic degrees of freedom (DoF). Here, we report a comprehensive study focusing on the phonons as well as intra-configurational excitations in double-perovskite Eu2ZnIrO6. Our studies reveal strong coupling of phonons with the underlying magnetic DoF reflected in the renormalization of the phonon self-energy parameters well above the spin-solid phase (TN ~ 12 K) till temperature as high as ~ 3TN, evidences broken spin rotational symmetry deep into the paramagnetic phase. In particular, all the observed first-order phonon modes show softening of varying degree below ~3TN, and low-frequency phonons become sharper, while the high-frequency phonons show broadening attributed to the additional available magnetic damping channels. We also observed a large number of high-energy modes, 39 in total, attributed to the electronic transitions between 4f-levels of the rare-earth Eu3+ ion and these modes shows anomalous temperature evolution as well as mixing of the crystal-field split levels attributed to the strong coupling of electronic and lattice DoF.


Introduction
A tremendous interest in iridates has been seen in the past few years due to a realization of the novel exotic quantum phases of matter ascribed to the strong spin-orbit coupling (SOC), electronic correlations, and their entanglement [1][2][3][4][5][6][7][8][9][10][11][12]. Double-perovskite iridium oxides of A2BIrO6 structure are of particular interest because of the complex quantum magnetic ground states in these materials. Their magnetic ground state is quite enigmatic, it strongly depends on the choice of A and B-site elements, that also provide an opportunity to tune the nature of magnetic interactions. It is anticipated that such tuning of these interactions may also give rise to the much sought after quantum spin-liquid state in these double-perovskites, considered to be the holy grail of the condensed matter physics. Despite advances in an understanding of different magnetic interactions, such as Heisenberg and Kitaev-type, in these systems with the substitution of 3d-transition metal elements (magnetic and nonmagnetic) of varying ionic radii at B-site i.e.
Where the A-site is occupied by an alkaline and/or rare-earth La-element; detailed investigations of complex magnetic behavior with other rare-earth elements at A-site are not much explored such as Ln2CoIrO6 (Ln = Eu, Tb and Ho) and Ln2ZnIrO6 (Ln = Sm, Nd) [21][22][23].
Eu2ZnIrO6 adopt a monoclinic double-perovskite structure with the space group P21/n (no. 14) [24]. Eu 3+ (4f 6 , S = 3, L = 3) is at eightfold coordinated A-site, whereas B-site is occupied by Zn 2+ (3d 10 ; spin S = 0), and Ir 4+ state with 5d 5 in psudo-spin Jeff = ½ occupies two different crystallographic sites in the corner-sharing octahedral environment. Eu2ZnIrO6 shows a ferromagnetic and/or canted antiferromagnetic like transition at TN ~ 12 K attributed to the ordering of spin associated with the Ir sublattice as Ir 4+ is the only magnetic ion in this compound, and Eu 3+ is a Van Vleck paramagnetic ion [24][25]. In Eu2ZnIrO6, Eu 3+ ion occupies 3 C1 low symmetry site, and therefore provides an opportunity to probe electronic degrees of freedom via potential electronic transitions between the different 4f-level multiplets. In these systems, magnetic, electronic and lattice degrees of freedom attract great deal of attention as they are believed to be main controlling factors central to their underlying physics. A very important aim in the study of Ir based double-perovskite is to elucidate their ground states and the contribution of different DoF. Here, we have undertaken such a study focusing on the intricate coupling between these three DoF, i.e., lattice, magnetic and electronic. We have used inelastic light (Raman) scattering as a tool to understand the coupling between these quasi-particle excitations.
Raman spectroscopy is a sensitive and effective technique for probing the underlying lattice, magnetic and electronic degrees of freedom, and is extensively used to unravel the coupling between a range of quasiparticles in solids such as magnons, orbitons, excitons and phonons [23,[26][27][28][29][30][31][32]. It is basically a photon-in-photon-out process in which an incident photon of energy i  is inelastically scattered

Experimental and computational details
The synthesis process and detailed characterization of these high-quality polycrystalline Eu2ZnIrO6 samples are describe in detail in ref. [24]. Inelastic light (Raman) scattering measurements were performed via Raman spectrometer (LabRAM HR-Evolution) in quasi-back scattering configuration using a linearly polarized laser of 532 nm (2.33 eV) and 633 nm (1.96 eV) wavelength (energy) at low power, less than ~ 1mW, to avoid local heating effects. A 50x long working distance objective was used to focus the laser on the sample surface and the spectra was dispersed through grating with density of 600 lines/mm. A Peltier cooled charged-coupled device detector was employed to collect the dispersed light. Temperature variation was done 5 from 4-330 K using a continuous He-flow closed-cycle cryostat (Montana instruments) with a temperature accuracy of ± 0.1 K or even better temperature accuracy.
The calculations of zone-centered phonons were done using plane-wave approach implemented in Quantum Espresso [33]. All the calculations were carried out using generalized gradient approximation with Perdew-Burke-Ernzerhof as an exchange correlation function. The plane wave cutoff energy and charge density cutoff were set to 60 and 280 Ry, respectively.
Dynamical matrix and eigen vectors were determined using density functional perturbation theory [34]. The numerical integration over the Brillouin-zone was done with a 4 x 4 x 4 k-point sampling mesh in the Monkhorst-Pack grid [35]. In our calculations, we used fully relaxed ionic positions with experimental lattice parameters given in ref. [24].

A. Raman-scattering and lattice-dynamics calculations
Eu2ZnIrO6 crystals have double-perovskite monoclinic structure belonging to space group P21/n (No. 14) [24]. The factor-group analysis predicts a total of 60 modes in the irreducible representation, out of which 24 are Raman active and 36 correspond to infrared active modes (see Table-I  In order to decipher the symmetry and associated eigen vectors (atomic displacements) of different vibrational modes, we performed Brillouin zone-centered (0, 0, 0) = lattice-dynamics calculations using density functional theory (DFT). We note that our calculated zone-centered 6 phonon frequencies are in very good agreement with the experimentally observed frequencies of the modes below 750 cm -1 at 4 K, labeled as S1-S17 (see Fig. 1(b) and Table-II). Based on our first-principle calculations, we have assigned these modes below ~ 750 cm −1 , i.e. modes S1-S17, as the first-order phonon modes, while high-frequency modes in the frequency range of 1000-1400 cm -1 , i.e. S18-S20, have been assigned as second-order phonon modes. The difference between calculated and experimental phonon frequency value at 4 K is determined via absolute ,N is the number of Raman active modes (17 here), and r %  is ~ 1.5% (obtained value). Furthermore, the symmetry assignment of different phonon modes is done in accordance to our first principle lattice-dynamics calculations shown in Table-II. We notice from the visualization of phonon eigen vectors (see Fig. 2) that the low-frequency phonon modes S1-S5 arise due to the displacement of Eu-atoms, however the high-frequency phonon modes S6-S17 comprise of the bending and stretching vibrations of Zn/Ir-O bonds associated with corner-sharing Zn/IrO6 octahedra. Furthermore, we also calculated full phonon dispersion and phonon density of states of Eu2ZnIrO6.  while the high-frequency density of states are associated with the oxygen-atoms. Apart from the first-order phonon modes, we observed a large number of additional modes at high-energy in the spectral range of 1400-4900 cm -1 , labeled as P1-P39 shown in Figure 1(c), attributed to the intraconfigurational transition modes of 4f levels of Eu 3+ ions, the detailed discussion is presented in sections C and D.

B. Temperature dependence of the phonon modes
In this section, we will be focusing on understanding the temperature evolution of the first as well as second-order phonon modes. First, we will discuss the first-order phonon modes, i.e.
modes below ~ 750 cm -1 . As the temperature is lowered, conventionally the mode frequencies are expected to show blue-shift. At the same time, with decreasing temperature, anharmonic phonon-phonon interaction reduces which results in an increase of phonon lifetime (α 1/ FWHM) or a decrease of FWHM [14]. Figure 4 shows the temperature-dependence of frequency and linewidth of the first-order phonon modes. The following important observations can be made: (i) Low-frequency phonon mode S1 (~ 71 cm -1 ) exhibits nearly temperature independent behavior down to ~ 100 K, which gradually decreases with decrease in temperature up to ~ 40 K.
On further lowering the temperature, mode frequency exhibits a sharp decrease down to the lowest recorded temperature, while the corresponding damping constant shows normal behavior till ~ 100 K, below which it exhibits a change of slope. (ii) All other phonon modes, S2, S4, S5, S8, S10, S11 and S17 exhibit normal temperature dependence down to ~ 40 K, i.e. increase (decrease) in the mode frequency (linewidth) with decreasing temperature, and below ~ 40 K these modes exhibit renormalization, i.e. anomalous phonon softening. (iii) Linewidth of the high-frequency phonon modes S8, S10, S11 and S17 exhibits line-broadening at low-temperature below ~ 40 K, while minimal line-narrowing with change in slope is observed for the low-8 frequency phonon modes S2, S4 and S5 around this temperature. Appearance of anomaly in the phonon self-energy parameters within the spin-solid phase, i.e. long-range magnetically ordered phase, of magnetic materials is generally attributed to the entanglement of lattice with underlying spin degrees of freedom via strong spin-phonon coupling [26,28,[31][32]36].
Before discussing the origin of anomaly in the phonon self-energy parameters, we quantify the effect of anharmonicity on phonon modes by fitting the frequency and linewidth of the optical phonon modes with anharmonic phonon-phonon interaction model by considering the three ( 1 2 2 o   ==) and four-phonons  . 4). Considering the anharmonic phonon-phonon interaction picture, the linewidth is expected to be higher with the increasing phonon energy, as also observed here (see Fig. 4). Such increase in the linewidth with increased 9 phonon energy may be understood by the availability of more decay paths into phonons with equal and opposite wave-vector. The lowest observed phonon mode (S1) has only acoustic phonon mode channel to decay into and hence it is the sharpest, but higher energy modes will have the decay channel of the all the available lower energy phonons including the acoustic one, and hence are broader.
Phonon-phonon anharmonic model fits are in very good agreement with the observed change in the frequencies and linewidths in the temperature range of 40-330 K hinting that temperature evolution in this temperature is mainly governed by the lattice degrees of freedom. However, we noticed a pronounced deviation of the frequencies and linewidths of the phonon modes from the curve estimated by anharmonic phonon-phonon interaction model at low-temperature below ~ 40 K, indicating that this behavior can not be described within this anharmonic phonon-phonon interaction picture. For phonons in the spin-solid phase below TN, an additional decay channel is expected into pairs of magnons of equal and opposite wave-vector. These observed anomalies below ~ 40 K may be understood by taking into account the interaction of phonons with the underlying magnetic degrees of freedom. The change in the phonon frequency due to interaction of lattice with spin degrees of freedom is given as [38] ( ) ( ) .
corresponds to the bare phonon frequency, i.e. phonon frequency without spin-phonon coupling; where ij J is the super-exchange coupling integral) is spin-phonon coupling coefficient, which may be positive or negative, and is distinct for each phonon mode, is the scalar spin-spin correlation function. The spin-spin correlation function is related to the order-parameter ()  T , and within the mean field theory the .   ij SS is given as: ion), and the order-parameter (  ) is given as, Here, we have kept * suggests the presence of short-range spin-spin correlations deep into the paramagnetic phase, and hence broken spin rotational symmetry. We also observed that low-energy phonon modes show continuous linewidth narrowing below ~ 40 K, though with a change in slope; on the other hand, high-energy phonon modes show linewidth broadening below this temperature. This opposite behavior suggests that magnetic dispersion branches are comparable or higher in energy than these low-energy phonons, and hence these additional decay channels are mostly available only to the higher energy phonons to decay into, which showed linewidth broadening. We hope that our studies will inspire a detailed theoretical study of magnetic dispersion in these systems to quantitively understand these anomalies.
This section will discuss the origin of the high-frequency phonon modes (i.e. S18-S20) observed in the frequency range of 1000-1400 cm -1 . Considering the frequency range of these modes and the fact that these are much above the observed first-order phonon modes limiting frequency ( ~ 700 cm -1 ), these modes are attributed to the second-order phonon modes corresponding to the S15-S17 first-order modes. Their origin in the intraconfigurational transitions of Eu 3+ ion is ruled out as they are also observed at the same position (i.e. Raman shift) with different excitation laser i.e. 633 nm. Second-order phonon modes are generally broader than their first-order counterparts because they involve the phonons over the entire Brillouin Zone with major contribution from the region of higher density of states. Specifically, the breadth of a secondorder Raman band is governed by the dispersion of the first-order phonon modes; and also, their peak frequencies are not necessarily double of those of the first-order phonons at the gamma point. We have assigned the second-order modes S18-S20 as overtone of first-order modes S15-S17, respectively. We note that, these second-order modes may also be assigned as possible combination of the different first-order modes, e.g. S20 may be assigned as combination of S16 and S17. Figure 5 shows the temperature evolution of the peak frequency, FWHM and intensity of the prominent second-order mode i.e. S20. The following observations can be made: ) in the entire temperature range, and the linewidth at low-temperature is more than double of the linewidth of mode S17.

C. Intraconfigurational modes of Eu 3+ in Eu2ZnIrO6
Apart from the first and second-order phonon modes, we observed a large number of modes in the spectral range of 1400-4900 cm -1 (see Figs The high-energy spectrum is fitted with a sum of Lorentzian functions (see Fig. 1(c)) and extracted peak position and linewidth of the prominent modes at 4 K are tabulated in Table-III Table- Table-IV. The majority of the observed modes are shifted towards higher energy with decreasing temperature and are fit well with the eq n . 4 suggesting that the associated energy levels move downward with increasing temperature. The transition modes following the temperature-dependent behavior similar to the phonons (blueshift with decreasing temperature) unambiguously signals their phonon mediated nature. The value of ion-phonon parameter (Wi) is negative for these modes, however this value is positive for other modes showing red-shift with decreasing temperature (see Table-IV). The red-shift of some of the observed modes with decreasing temperature may be understood by so-called pushing effect or fast lowering of the terminal levels [51]. A similar red-shift has also been reported for other rare-earth elements [23,[51][52][53]. It has been associated with the fact that terminal levels lower faster than other levels with increasing temperature, therefore transition from higher level to these terminal levels of any set of multiplets will result in the red-shift. More theoretical work is required to understand these red-shifts and their coupling with other quasiparticles.

Summary and conclusions
To summarize, we report the detailed lattice-dynamics study of double-perovskite Eu2ZnIrO6 via Raman scattering as a function of temperature and density functional theory-based calculations.
We find significant phonon softening and anomalous linewidth narrowing/broadening of the