Kramer doublets, phonons, crystal-field excitations and their coupling in Nd2ZnIrO6

We report comprehensive Raman-scattering measurements on a single crystal of double-perovskite Nd2ZnIrO6 in temperature range of 4-330 K, and spanning a broad spectral range from 20 cm-1 to 5500 cm-1. The paper focuses on lattice vibrations and electronic transitions involving Kramers doublets of the rare-earth Nd3+ ion with local C1 site symmetry. Temperature evolution of these quasi-particle excitations have allowed us to ascertain the intricate coupling between lattice and electronic degrees of freedom in Nd2ZnIrO6. Strong coupling between phonons and crystal-field excitation is observed via renormalization of the self-energy parameter of the phonons i.e. peak frequency and line-width. The phonon frequency shows abrupt hardening and line-width narrowing below ~ 100 K for the majority of the observed first-order phonons. We observed splitting of the lowest Kramers doublets of ground state (4I9/2) multiplets i.e. lifting of the Kramers degeneracy, prominently at low-temperature (below ~ 100 K), attributed to the Nd-Nd/Ir exchange interactions and the intricate coupling with the lattice degrees of freedom. The observed splitting is of the order of ~ 2-3 meV and is consistent with the estimated value. We also observed a large number of high-energy modes, 46 in total, attributed to the intra-configurational transitions between 4f3 levels of Nd3+ coupled to the phonons reflected in their anomalous temperature evolution.


Introduction
Physics of correlated electron systems associated with 5d transition-metal oxides, especially Iroxides, has drawn considerable research interest in recent years owing to the possible formation of quantum-spin-liquids, Mott-insulators, unconventional superconductors and Weyl-semimetals [1][2][3][4][5][6][7][8][9][10]. Iridates provide an interesting interplay between strong spin-orbit coupling and electron correlation due to their comparable strength along with exotic quasi-particle excitations such as orbitons [11]. Double perovskite iridium-based materials of A2BIrO6 crystal structure are interesting as well as challenging due to the freedom of tuning the quantum magnetic ground state with the choice of different magnetic and non-magnetic ions on A and B crystallographic sites. So far, a majority of the studies have focused on tuning the magnetic exchange interactions by replacing B-sites with magnetic (i.e. Mn, Fe, Co, Ni and Cu) and non-magnetic (i.e. Y, Mg and Zn) elements [12][13][14][15][16][17][18][19][20]. The evolution of exotic quantum magnetic properties with the choice of Asite element in these materials is not extensively explored [21][22]. The substitution on A-site with different lanthanide rare-earth elements may provide another degree of freedom to give insight into the complex ground state in these 5d-materials. This replacement may cause substantial local structural changes (i.e. change of bond distance Ir-O, bond angle O-Ir-O, and rotation and/or tilting of IrO6 octahedra) within the unit cell, and may result in significant modulation of the strength of exchange interactions between nearest-neighbors. Depending on the ionic-radius of A-site ions, different quantum magnetic spin interactions such as Heisenberg and Kitaev type may be realized in double pervoskite iridates [23][24][25]. Another important aspect of the presence of rare-earth ion in these systems is the crystal field excitations (CFE) originating from the splitting of rare-earth ion multiplets with specific local site symmetry owing to the surrounding ions static field. Probing these electronic excitations are very important because CFE as a function of temperature may 3 provide crucial information about the nature of underlying ground state ordering and also provide the avenues to probe the coupling with lattice degrees of freedom [26][27][28][29].
The magnetic phase of Nd2ZnIrO6 is rather mysterious. It shows an antiferromagnetic ordering at ~ 17 K attributed to the ordering of Nd 3+ ions with a small contribution from Ir sub-lattice, along with a broad transition at 8 K. However, under weak magnetic field, a complex magnetic phase is reported with multiple magnetic transitions [22] attributed to the interplay between Nd and Ir magnetic sub-lattices. Nd 3+ ion here inhabits an eight-fold coordinated A-site with C1 local site symmetry that shows there is no symmetry element except identity. The sensitivity of the Nd 3+ crystal-field levels to both local electric and magnetic fields makes Raman scattering by these CF levels a valuable probe for exploring the possible underlying interactions responsible for their ground state. Measurements of CFE are also important to understand various thermo-dynamical properties and accurately determine the generalized phonon density of states. Their measurements via Raman is complementary to complex neutron-scattering measurements, where one requires a big single-crystal. Here, we present a detailed study of a single-crystal of double-perovskite Nd2ZnIrO6 focusing on the phonons, CFE, and the coupling between them. We note that not much attention has been paid for understanding the role of phonons and the coupling with other quasiparticle excitations in these Ir-based double-perovskite systems. We have performed a detailed lattice-dynamic study of Nd2ZnIrO6 in the temperature range of 4 K to 330 K, and polarizationdependent Raman-scattering measurements along with the zone-centered phonon mode frequency calculations using first-principle based density functional theory (DFT). Our measurements reveal strong coupling between phonons and CFE, and evidence the lifting of Kramer's degeneracy of Nd 3+ ground state multiplets at low-temperature attributed to the Nd-Nd and Nd-Ir exchange interactions.

Experimental and computational details
The inelastic light (Raman) scattering measurements were performed on a single-crystal of Nd2ZnIrO6 in back scattering configuration via Raman spectrometer (LabRAM HR-Evolution).
We have employed 532 nm (2.33 eV) and 633 nm (1.96 eV) lasers as an excitation probe for spectral excitation. The laser was focused on the sample surface using 50x long working distance objective lens. A Peltier-cooled charge-coupled device (CCD) detector was used to collect the scattered light. Laser power was limited (≤ 1mW) to avoid any local heating effects. Temperaturedependent measurements were performed with a closed-cycle He-flow cryostat (Montana Instruments), where sample is mounted on top of the cold finger inside a chamber, which was evacuated to 1.0x10 -6 Torr, in the temperature range of 4 K to 330 K with ± 0.1 K or better temperature accuracy.
Density functional theory-based calculations were done to have insight about phonon dynamics using plane-wave approach implemented in QUANTUM ESPRESSO [30]. The fully-relativistic Perdew-Burke-Ernzerhof (PBE) was chosen as an exchange correlation functional within generalized gradient approximation to carry out these calculations. A plane wave cut-off energy and charge density cut-off was set to 60 Ry and 280 Ry, respectively. The numerical integration over the Brillouin-zone was done with a 4 x 4 x 4 k-point sampling mesh in the Monkhorst Pack [31]. Dynamical matrix and zone-center phonon frequencies were calculated using density functional perturbation theory, taking spin-orbit coupling without spin-polarization into account [32]. All the calculations were performed with experimental lattice parameters reported in ref. 22.

A. Raman-scattering and Zone-centered calculated phonon frequencies
Nd2ZnIrO6 crystallizes in a monoclinic double-perovskite structure belonging to the P21/n space group (No. 14) and C2h point group [22]. Within this structure the factor-group analysis predicts a total of 60 modes in the irreducible representation, out of which 24 are Raman active and 36 are infrared active (see Table-I for details). The Raman spectra of Nd2ZnIrO6 single-crystals were excited with two different incident photon energies 633 nm (1.96 eV) and 532 nm (2.33 eV). Figure   1 shows the Raman spectra of Nd2ZnIrO6 excited with 633 nm laser at 4 K, while the insets shows its temperature evolution and comparison with 532 nm at 4 K and 330 K. We note that the Raman spectra when excited with 532 nm exhibits extra features in the spectral range of 50-800 cm -1 and 1700-5500 cm -1 , labeled as R1-R7 (see inset a2 of Fig. 1) and Q1-Q46 (see Figure 6 (a and b)), respectively. A detailed discussion about these modes is discussed in later sections. First, we will focus on the first-order phonon modes shown in Figure 1. We notice a total of 23 phonon modes in the spectrum excited with 633 nm laser, labeled as P1-P23. The data is fitted with a sum of Lorentzian functions to obtain the spectral parameters such as phonon mode frequency (  ), linewidth (  ) and the integrated intensity.
In order to have insight on the specific vibration patterns of the observed phonon modes, we carried out zone-centered lattice dynamics calculations using density functional perturbation theory. The calculated mode frequencies together with atomic displacements (vibrations) associated with particular phonon modes of Nd2ZnIrO6 are listed in Table-II. The calculated phonon modes frequencies are found to be in very good agreement with the experimentally observed values. Figure 2 illustrate the schematic representations of the calculated eigen-vectors of the corresponding phonon modes. The evaluation of these eigen-vectors reveals that the lowfrequency phonon modes P1-P3, P5 are mainly attributed to the atomic displacement of Nd-atoms, while P4 is composed of the displacement of Nd/Zn/Ir. The high-frequency phonon modes P6 to P23 are corresponds to the stretching and bending of the Zn/Ir-O bonds in the Zn/IrO6 octahedra. 6 The phonon modes assignment is done in accordance to our density functional theory calculations and polarization-dependent Raman measurements. Figure 3 shows the temperature-dependence of the frequency and line-width of the first-order phonon modes of Nd2ZnIrO6 excited with 633 nm laser. The following important observation can be made: (i) Low-energy phonon mode P1 (~ 100 cm -1 ) exhibit phonon softening below ~ 100 K, however no anomaly is observed in the line-width of this mode in the entire temperature range.

B. Temperature and polarization-dependence of the first-order phonon modes
(ii) Interestingly, the phonon modes P3-P4, P7-P8, P10-P11, P15 and P21-P22 show significant frequency hardening with a decrease in temperature below ~ 100 K down to the lowest recorded temperature, while P20 and P23 phonon modes display a slight softening. Additionally, a clear change in the line-widths of all these phonon modes is observed around ~ 100 K, i.e. anomalous line-narrowing below this temperature. Similar temperature-dependence is also observed for the frequency and line-width of phonon modes excited with 532 nm laser (not shown here). We note that the anomalies are more pronounced for the low-frequency phonon modes. The observed anomaly in the phonon modes frequencies and line-widths at low-temperature clearly indicate strong interaction of the phonons with other degrees of freedom. Long-range magnetic ordering in this system is reported at ~ 17 K [22], however the phonon anomalies are observed at temperature as high as ~ 100 K, well above the magnetic transition temperature. This observation suggests its origin from other degrees of freedom in the observed phonons anomalies than magnetic ones, such as electronic degrees of freedom. Here, a coupling to crystal-field multiplets may be at work.
To understand the temperature-dependence of the first-order phonon modes, we have fitted the phonon mode frequency and line-width from ~ 100 K to 330 K with an anharmonic phononphonon interaction model including both three and four-phonons decay channels given as [33]: where 0  and 0  are the mode frequency and line-width at T = 0 K, respectively,  Table-II. The expected temperature variation of mode frequency and line-width due to anharmonicity is shown as solid red and blue lines (see Fig. 3). Within the anharmonic interaction picture phonon modes are expected to become sharper and mode frequency gradually increase with decreasing temperature.
Notably, at low-temperature below ~ 100 K the phonon mode self-energy parameters, i.e. mode frequencies and line-widths, exhibit significant deviation from the curve estimated by anharmonic phonon-phonon interaction model. Majority of the observed phonon mode frequencies harden anomalously below ~ 100 K, except mode P1, P20 and P23. Interestingly, the magnitude of the phonon frequency hardening is observed to be more pronounced for the low-frequency phonon modes (below 300 cm -1 ). In general, the magnitude of anomalous modulation of phonon frequencies are usually attributed to the interaction strength of a particular phonon mode with, e.g., spin degrees of freedom. Hence, in a long-range magnetic phase the phonon renormalization mainly occurs due to the strong coupling of lattice with spin degrees of freedom. The effect of magnetic degrees of freedom is expected to be minimal if not zero in this case because TN (~ 17 K) is much lower than the temperature (~ 100 K) where anomalies are observed. In this scenario, the pronounced self-energy renormalization of the phonon modes implies a strong coupling of lattice with electronic degrees of freedom via crystal-field excitations associated 4f-levels of Nd 3+ ion. Indeed, we found signature of strong coupling of these two degrees of freedom where CFE excitations, which are also in the same energy range and have similar symmetry, arising from 8 ground state ( 4 9/2 I ) multiplets of Nd 3+ ion at low-temperature and led to the strong renormalization of the phonon self-energy parameters.
To decipher the symmetry of the phonon modes, we also performed polarization-dependent measurements. Polarization-dependent measurements using Raman spectroscopy can be done  Table- . Here  o is an arbitrary angle from a-axis and is constant. Therefore, without the loss generality it may be chosen to be zero, giving rise to the expression for the Raman intensity as The fitted curves are in very good agreement with the experimental angle variation of the mode intensities. Using the polarization-dependence of the phonon modes and our DFT based calculations, we have assigned the symmetry of the observed phonon modes (see Table-II).
Symmetry of the phonon modes extracted from our polarization-dependent measurement is mapped nicely with the mode symmetries calculated using the first principle DFT based calculations.

C. Crystal-field excitations and coupling with phonons
In addition to the phonon modes several additional modes are observed in the spectrum excited with 532 nm (2.33 eV) laser. First, we will discuss the modes observed in the low-frequency range below ~ 500 cm -1 , labeled as R1-R7 (see Fig. 5(c) and inset a2 of Fig. 1), which appear in low-   Fig. 5(b)), which is typical for Ramanscattering associated with crystal-field transitions (because the ground state population increases with decreasing temperature) and shows temperature evolution opposite to those of phonons (see right side of Fig. 5(b) showing intensity for modes P1, P3 and P4). We also did the polarizationdependence of these modes at low-temperature (see Fig. 5 (d) and inset of Fig. 5(c)). The polarization-dependence of these modes reflect their quasi-isotropic nature and are quite different from the observed phonon modes polarization behavior, confirming the non-phononic nature of these modes. From the symmetry analysis, CF levels of the Nd 3+ in ground state (i.e. 4 9/2 I ) multiplets split into five Kramer's doublets. Since Nd 3+ has a C1 local site symmetry, all the levels have A symmetry and transition between them are expected to be active in all polarizations [39].
Therefore, the observation of quasi-isotropic nature of the polarization-dependent behavior of these CF modes is in-line as predicted by the symmetry analysis. We note that these lower-energy 11 crystal-field modes appear only at low temperature (below ~ 200 K). The Raman cross-section for the CFE is often weak and these have been observed generally with the strong coupling with nearby Raman active phonon modes via strong electron-phonon interaction. Two quasi-particle excitations of similar energy and symmetry are expected to couple strongly [39], therefore a phonon mode and CFE of same symmetry and similar energy are also expected to couple strongly.
As discussed in the previous section above, renormalization of the phonon modes at lowtemperature (~ 100 K) is very strong (see Fig. 3) and interestingly renormalization effect are more pronounced for low-energy phonon modes (e.g. P1, P3, P8, P10 and P11; all have g A symmetry).
Energy range of these phonon modes (maximum ~ 400 cm -1 ) is similar to those of CFE (maximum ~ 450 cm -1 ) and both these excitations are of similar symmetry. Therefore, our observation of renormalization of the phonon modes at low-temperature is attributed to the strong coupling with these crystal-field excitations.
Another interesting observation is that the number of possible transitions within the ground state multiplets at low-temperature are expected to be only four as shown in the inset of Figure 5 (c), however seven modes are observed (R1-R7) i.e. three doublet (R1-R2, R3-R4, R5-R6) and mode R7. At low-temperature we observed four strong modes i.e. R2, R4, R6 and R7 and with further decrease in temperature three weak shoulder modes namely R1, R3 and R5 also started gaining spectral weight. Frequency difference between these doublets is ~ 2-3 meV (~ 20-25 cm -1 ) and it increases with decrease in temperature (see inset in top panel of Fig.5(a)). The observed weak modes i.e. R1, R3 and R5 may arise from the splitting of the ground state Kramer's doublet by the Nd-Nd and Nd-Ir exchange interactions. Interestingly, with increasing temperature the overall intensity of the higher-frequency doublet components ( i.e. R2, R4 and R6 ) decreases relative to the lower-frequency ones (i.e. R1, R3 and R5) (see Fig. 5 (c)). Also, discontinuities in the intensity follow the long-range magnetic ordering (see inset in top and middle panel in Fig.5 (b)) associated with the two interpenetrating AFM sub-lattices associated with the Nd and Ir moments ordering.
This discontinuity is also visible in the doublet splitting (see inset in the top panel of Fig. 5 (a)).
Quantitatively, the origin of these discontinuities may be understood using the fact that splitting of the Kramer's doublets is due to Nd-Nd and Nd-Ir exchange interactions, as these exchange

D. Intra-configurational transitions and interaction with phonons
We observed a large number of sharp and strong modes, labeled as Q1-Q46, in the high-energy range spanning from 1700 to 5500 cm -1 , shown in Figure 6 (a, b) FS + are at ~ 17000 cm -1 , 14700 cm -1 , and 13800 cm -1 , respectively, [37][38] (see Fig. 6(c) for schematic of crystal-field splitting of these levels). Based on the comparative energy with different CF levels, modes from Q1-Q20 may be assigned to the transitions from higher lying doublets of ( increase in the line-widths, say as one goes from mode Q6 to Q20; and from Q22 to Q38 (see Table-III for the list of prominent modes line-width with this increasing trend), attributed to the strong electron-phonon interaction. We know that a narrow line-width is associated with the electronic transitions from the lowest multiplet of any crystal-field level to the lowest level of other multiplets of CF level, and it increases for transitions to the higher multiplets of the same CF level [41]. The underlying mechanism for such a broadening of the line-width is due to the possible relaxation to the available lower-energy levels with the spontaneous emission of phonons.
where, 0i E is the energy at 0 K, W is the ion-phonon coupling constant, D  is the Debye phonon energy, and D  is the Debye temperature taken as ~ 950 K from our density of phonon calculations. We have fitted the frequency of the prominent mode using the above expression, see solid lines in Figure 7 (a, b) and the fitting parameters are given in Table-IV. Fitting is in good agreement in the temperature range of 330 K to ~ 100 K, below 100 K it starts diverging from the expected behavior. The majority of these modes show a normal temperature dependence of the frequencies, i.e. increase in energy with decreasing temperature, until ~ 100 K. The temperature evolution may be gauged from the fact that these electronic transitions are mediated by phonons, and they are expected to show similar temperature-dependence as that of phonons. 15 From the fitting, the ion-phonon interaction parameter ( i W ) is negative for these modes (see Table-IV). On the other hand, the temperature-dependence of mode Q28 and Q29 is anomalous i.e. peak frequency decrease with decreasing temperature, also reflected in the positive value of i W for mode Q29. We note that such anomalous temperature evolution has also been observed for different systems [42][43][44].

Summary and conclusion
In conclusion, anomalous renormalization of the first-order phonon modes below ~ 100 K is attributed to the strong coupling between lattice degrees of freedom and crystal-field excitations.
The phonon mode frequencies and their symmetries are estimated using first-principle density functional theory-based calculations, which are in very good agreement with our polarization dependent Raman measurements. We observed splitting of ground state Kramer's doublets attributed to the exchange interaction between the Nd-Nd/Nd-Ir, and the observed doublet splitting is in very good agreement with the estimated values. Our results shed crucial light on the role of A-site rare-earth element in these iridium based double-perovskite materials reflected via coupled lattice and electronic degrees of freedom, and believe that our results will prove to be an important step in understanding the exotic ground state of these systems by considering electronic and phononic degrees of freedom on equal footing. We hope that our detailed experimental studies reveling a large number of intra-configurational transitions between CF-levels of Nd 3+ at low-site symmetry also provide a good starting point to calculate the energy of all possible CF-levels in these important class of double-perovskite systems using theoretical models.