Unconventional spin-phonon coupling via the Dzyaloshinskii–Moriya interaction

Spin-phonon coupling (SPC) plays a critical role in numerous intriguing phenomena of transition metal oxides (TMOs). In 3d and 4d TMOs, the coupling between spin and lattice degrees of freedom is known to originate from the exchange interaction. On the other hand, the origin of SPC in 5d TMOs remains to be elucidated. To address this issue, we measured the phonon spectra of the 5d pyrochlore iridate Y2Ir2O7 using optical spectroscopy. Three infrared-active phonons soften below the Néel temperature of TN ≈ 170 K, indicating the existence of strong SPC. Simulations using density functional theory showed that the coupling is closely related to the Ir–O–Ir bond angle. A tight-binding model analysis reveals that this SPC is mainly mediated by the Dzyaloshinskii–Moriya interaction rather than the usual exchange interaction. We suggest that such unconventional SPC may be realized in other 5d TMOs with non-collinear magnetic order.


INTRODUCTION
The coupling between various degrees of freedom (spin, charge, lattice, and orbital) is an important paradigm in condensed matter physics. Such couplings are known to give rise to novel ground states and various emergent phenomena, [1][2][3][4][5] including high T C superconductivity, 2,3 colossal magnetoresistance, 4 etc. Recently, there has been increasing interest in spin-phonon coupling (SPC) within the context of multiferroics and spintronics. [5][6][7][8][9] In multiferroic materials, SPC has been invoked to explain intriguing phenomena such as the thermal Hall effect 8 while it also determines the spin relaxation time in spintronics applications. 9 Moreover, SPC presents the opportunity to engineer new functionality in transition metal oxides (TMOs). For example, it is possible to stabilize a specific magnetic ground state with epitaxial strain 10 or generate magnetoelastic effects by changing the magnetic easy-axis. 7 In this context, it has become increasingly important to develop a detailed understanding of SPC.
In general, we can describe the physics of most spin-related phenomena in solids using the spin Hamiltonian (H spin ). This can be written as a sum of the isotropic exchange (IE) interaction, the Dzyaloshinskii-Moriya (DM) interaction, the anisotropic exchange (AE) interaction, and the single-ion anisotropy (SIA): [11][12][13] where J IE , D ij , and Γ ij are the coefficients of the IE, DM, and AE interactions, respectively. A and n i are the coefficient of the SIA and the direction vector of corresponding easy-axis. S i and S j are the spins at the nearest neighboring i th and j th sites. 14,15 Spin interactions are mediated by the Coulomb interaction of electrons at neighboring sites 16 and are highly dependent on the positions of the neighboring ions. Consequently, J IE , D ij , Γ ij and/or A terms can be dynamically modulated by a lattice vibration, resulting in SPC and a renormalization of the phonon frequencies.
Among 5d TMOs, pyrochlore iridates are good candidates to search for new types of SPC. First, they have an intriguing ground state, called J eff = 1/2 state, 15,34 due to strong SOC. Second, pyrochlore iridates host an intriguing form of antiferromagnetism, namely the all-in-all-out (AIAO) magnetic order. 32,35 As shown in Fig. 1a, all spins of one (neighboring) tetrahedron point inward (outward). 36 This non-collinear spin configuration results in a large 〈S i × S j 〉 value in equation (1). Significant dynamic modulation of the DM interaction can therefore lead to a renormalization of the phonon frequencies which cannot be fully explained by the IE interaction alone.
To investigate SPC with the AIAO magnetic order, we focused on polycrystalline samples of Y 2 Ir 2 O 7 . In Fig. 1b, the magnetic susceptibility of this compound clearly indicates antiferromagnetic ordering at Néel temperature T N ≈ 170 K. We verified the crystal structure of our sample using X-ray diffraction (XRD), as shown in Fig. 1c. The good agreement between experimental XRD results and the Rietveld refinement fitting confirmed the pyrochlore structure of our sample.
The phonon renormalization due to SPC can be directly measured by infrared (IR) spectroscopy. 11 In general, SPC phenomena have been widely investigated by measuring anomalies in spin wave dispersion with either neutron or X-ray scattering experiments. 37,38 However, in Y 2 Ir 2 O 7 the constituent Ir ions strongly absorb neutrons, making neutron scattering experiments difficult. Resonant inelastic X-ray scattering experiments could be another option, but the precise analysis of anomalies due to SPC is complicated by a comparatively poor energy resolution (i.e.,~10 meV). 39 In contrast, the high energy resolution (~0.5 meV) of IR spectroscopy allows us to investigate the details of SPC by precise measurement of the phonon frequency shifts across the T N .
Here, we report evidence for unconventional SPC via the DM interaction in Y 2 Ir 2 O 7 . To investigate DM-derived SPC, we monitored the temperature (T)-dependent phonon spectra in Y 2 Ir 2 O 7 by infrared spectroscopy. Three IR-active phonons soften and sharpen below T N , implying the presence of strong SPC. Based on first-principles calculations and tight-binding model analysis, we show that the observed phonon frequency shifts originate from the modulation of the Ir-O-Ir bond angle and establish that the phonon renormalization in Y 2 Ir 2 O 7 is predominantly driven by the DM interaction.

RESULTS
Phonon spectra measurement and Fano-Lorentz model analysis In the present work, we investigated the IR-active phonons of a Y 2 Ir 2 O 7 polycrystalline sample. The photon energy range was 6 ≤ ℏω ≤ 5,000 meV and T ranged from 10 to 300 K. As presented in Fig. 2a, the σ 1 (ω) spectra show seven IR-active phonons in Y 2 Ir 2 O 7 39-41 below 90 meV, as expected for the pyrochlore structure. 40,41 The experimental σ 1 (ω) spectra also showed several weak peaks, possibly due to impurities, 42 but they did not influence our phonon analysis of Y 2 Ir 2 O 7 . For later discussion, we label the seven strong phonon modes as Ph1 to Ph7 in order of increasing frequency, ω.
Three of the IR-active phonons exhibit an anomalous Tdependence. Figure 2b shows the T-dependent evolution of each phonon. The blue arrows show that Ph1, Ph2, Ph3, and Ph7 blueshift (i.e., harden) monotonically with decreasing T. On the other hand, the red arrows indicate that Ph4, Ph5, and Ph6 redshift (i.e., soften) abruptly. To perform a more quantitative analysis, we fitted the σ 1 (ω) spectra with a Fano-Lorentz model. A comparison with the phonon-phonon scattering theory can provide an insight into the phonon renormalization of Y 2 Ir 2 O 7 . The red lines in Fig. 3 indicate the theoretical phonon frequencies, ω 0 th , expected based on the simple phonon-phonon scattering theory. 27,43 The ω 0 (T) of Ph7 is well-described by the theory. On the other hand, the ω 0 (T) of Ph4, Ph5, and Ph6 show abrupt redshifts at T N ≈ 170 K that are not captured by the theory. For later discussion, we define the frequency difference, Δω 0 (T) ≡ ω 0 th (T) -ω 0 (T) at T = 10 K. In addition, the Γ(T) of Ph4, Ph5, and Ph6 also show kink-like features near T N , as shown in the lower panels of Fig. 3a-c. The phonon linewidth, the inverse of the lifetime, can become smaller in the ordered state which is away from the phase transition. However, earlier XRD and neutron diffraction experiments showed that AIAO magnetic ordering in pyrochlore iridates is not accompanied by a structural transition. [44][45][46] Therefore, we can conclude that the anomalous T-dependence of Ph4, Ph5, and Ph6 should be coupled to the AIAO magnetic order.
Phonon analysis with density functional theory (DFT) DFT calculations were performed to determine the phonon frequencies and corresponding eigenmodes of Y 2 Ir 2 O 7 (see Methods). The AIAO spin configuration of the pyrochlore structure was used for the calculations. These spins are aligned along the local Ir [111] axis in the Ir tetrahedra. Figure 4a shows the spincanting angle θ spin ¼ cos À1 À 1 to the experimental values from 127.5°to 128.5°measured by neutron diffraction 44,48 and XRD, 49 respectively. As shown in Table  1, the DFT-calculated phonon frequencies agree well with the experimental values and therefore support the validity of our calculations.
To obtain further insight into the phonon renormalization, let us focus on the Ir-O-Ir bond. Note that O is the lightest element in Y 2 Ir 2 O 7 , so its vibration amplitude should be largest. In this context, the motion of the O ion between the neighboring Ir spins plays an important role in the electron hopping along the Ir-O-Ir path and in H spin. 13 The seven IR-active phonon modes can be classified into three categories based on the dominant motion of the O ions along the x o -, y o -, and z o -directions (see Supplementary Note 2). The classification is shown in Fig. 4b: Ir-O-Ir stretching (Ph1 and Ph7), out-of-plane bending (Ph2, Ph3, and Ph4), and inplane bending (Ph5 and Ph6) motion, respectively (see Based on the DFT results for the phonon eigenmodes, we estimated the change in the bond angle θ for each phonon mode. We assumed that the magnitude of the ionic displacements associated with each eigenmode is~0.01 Å. Then, we calculated the concomitant bond angle change, Δθ ≡ θ − θ 0 = θ − 126.9°. The calculated Δθ/θ 0 ratios are plotted as a bar graph in Fig. 5. Note that Δθ/θ 0 becomes the largest for Ph6. The experimental Δω 0 /ω 0 ratios are also plotted as black squares. We found that Δω 0 /ω 0 and Δθ/θ 0 have similar trends. This behavior suggests that the unusual phonon renormalization in Y 2 Ir 2 O 7 is closely related to the modulation of the Ir-O-Ir bond angle. For conventional SPC, the bond length is the most critical parameter for modulating the IE interaction. 27,29,50 Change in the bond length may affect the overlap integral significantly, resulting in a sizable change in J IE. 51 In Y 2 Ir 2 O 7 , however, the Ir-O bond length change Δl has only a slight effect on Δω 0 . We calculated the length changes Δl/l 0 associated with each phonon mode and found that they deviate markedly from the experimental results, especially for Ph4 and Ph7 (see Supplementary Note 4). Additionally, if Δl is indeed the critical parameter for the phonon renormalization, the stretching modes would be expected to show the largest frequency shift. However, the stretching mode Ph7 shows no frequency shift near T N . These observations indicate that Δl cannot play important roles in Δω 0 at T N .

Importance of the DM interaction for the phonon renormalization
To elucidate the microscopic origin of SPC in Y 2 Ir 2 O 7 , we meticulously examined the relation between Δω 0 and Δθ. The bond angle change induced by lattice vibration can affect the H spin in Y 2 Ir 2 O 7 , given in equation (1). Due to the strong SOC in pyrochlore iridates, the low energy magnetic degrees of freedom may be represented by pseudospins, J eff = 1/2. 34,52 Due to the 1/2 value of J eff , the SIA term becomes negligible. 53 Previous studies have also concluded that the AE interaction is at least an order of magnitude smaller than the other terms in H spin.
13,54-56 Consequently, for Y 2 Ir 2 O 7 , Δω 0 may be largely attributed to the IE and DM interactions: where I is the rotational inertia of the given phonon 57 (see Supplementary Note 5 for derivation). The positive or negative sign of Δω 0 determines whether the corresponding phonon softens or hardens.   To compare the contributions of the IE and DM interactions quantitatively, we considered a tight-binding model based on Omediated Ir-Ir hopping. 13 Previous tight-binding models for pyrochlore iridates assumed that Ir-O hopping could occur along the local axes. 13,[54][55][56]58 However, the actual lattice structure includes a strong trigonal distortion of the O octahedra even though the global lattice symmetry is maintained. This distortion moves the O ion position away from the local axes of Ir ions (i.e., along the z L and y L axes in the Fig. 6a). Figure 6a show the changes of Ir-O-Ir bond during the trigonal distortion. Note that when the Ir-O-Ir bond angle changes to θ 0 , the Ir-O hopping is no longer aligned to the local axes. With proper consideration of the ionic position changes, we analytically obtained the θ-dependent Ir-O-Ir hopping. Then, using the Slater-Koster tables, we derived the bond angle-dependent coefficients J IE (θ) and D(θ) (=‖D ij (θ)‖) in H spin 54-56,58,59 (see Supplementary Note 5 for derivation). To make a quantitative comparison between J IE (θ) and D(θ) for Y 2 Ir 2 O 7 , we need to anchor their values for at least one angle. Unfortunately, experimental values have not yet been reported for Y 2 Ir 2 O 7 . We therefore rely on a previous quantum chemistry calculation (QCC) study that computed J IE (θ) (= 18.8 meV) and D(θ) (= 2.6 meV) at the experimentally determined angle θ 0 QCC (= 130°). 60,61 Using these QCC values, we quantitatively determined J IE (θ) and D(θ), which are displayed in Fig. 6b.
Using the J IE (θ) and D(θ) values, we then estimated the IE and DM interaction energies E IE = J IEij (θ)〈S i •S j 〉 and E DM = D ij (θ)•〈S i × S j 〉. By taking the second derivatives, i.e., the ∂ 2 E DM /∂θ 2 and ∂ 2 E IE /∂θ 2 curves shown in Fig. 6c, we can compare both the IE and DM contributions to Δω 0 using equation (22). The results suggest ∂ 2 E DM /∂θ 2 ≈ 1.00 eV/radians 2 and ∂ 2 E IE /∂θ 2 ≈ −0.09 eV/radians 2 at θ = 130°. The contribution to phonon renormalization of the DM interaction (∂ 2 E DM /∂θ 2 ) is therefore about an order of magnitude greater than that of the IE interaction (∂ 2 E IE /∂θ 2 ). The dominance of the DM interaction is robust even when the anchoring values of J IE (θ) and D(θ) were varied significantly. The DM interaction therefore plays a key role in the unconventional SPC observed in Y 2 Ir 2 O 7 .

DISCUSSION
We now discuss the importance of the bond angle, as this determines the dominant SPC mechanism. For the IE interaction, we can empirically write J IE θ ð Þ ¼ J 90 sin 2 θ þ J 180 cos 2 θ, where J 90°a nd J 180°r epresent the IE interaction terms from the bond angles 90°and 180°, respectively. 62 Using a Taylor expansion, J IE (θ) becomes an approximately linear function of θ around 135°. And as a result, the SPC due to the IE interaction (i.e., ∂ 2 E IE /∂θ 2 ) is strongly suppressed with θ~135°. In contrast, the DM interaction is expected to vary as sin(2θ), 63 meaning ∂ 2 E DM /∂θ 2 is zero at 180°a nd a maximum near 135°. Therefore, the DM interaction can play a dominant role in driving the SPC when θ is close to 135°.
To obtain a better perspective, we now compare the phonon renormalization of Y 2 Ir 2 O 7 to that of two other 5d TMOs: NaOsO 3 and Ca 2 Os 2 O 7 . The relevant physical parameters are summarized in Table 2. 11,50,60,61,64,65 Recently, Calder et al. reported that the breathing phonon mode of NaOsO 3 has Δω 0~5 .0 meV, the largest observed in solids. 50 This large renormalization was attributed to the strong modulation of the IE interaction by the Os-O-Os bond stretching motion. Indeed, NaOsO 3 has a distorted perovskite structure and θ is close to 180°. Following our earlier arguments regarding θ, the IE interaction is expected to be the leading contributor to the SPC. The SPC mechanism in NaOsO 3 may therefore be distinguished from that of Y 2 Ir 2 O 7 due to different bond angles in the two compounds.
We now turn to the SPC occurring in Cd 2 Os 2 O 7 , a compound that shares the same crystal structure and magnetic ground state     11 and attributed it to the SPC via the SIA. The SIA energy can change rapidly with the trigonal distortion of the OsO 6 octahedra and so is especially sensitive to phonons that modulate this distortion. 11,65 In contrast, in Y 2 Ir 2 O 7 , we can neglect the SIA term because of J eff = 1/2, as shown in equation (22). It is also of interest to note that, despite their structural and magnetic similarities, Cd 2 Os 2 O 7 and Y 2 Ir 2 O 7 reveal opposite signs of Δω 0 due to the different origins of the SPC.
In summary, we have established DM-mediated strong SPC in Y 2 Ir 2 O 7 using both IR spectroscopy and first-principles calculations. Remarkable variations in the phonon parameters (frequency and linewidth) were observed near T N . These changes can be understood in terms of the change in Ir-O-Ir bond angle induced by the relevant phonon. We also explained that the dynamic modulation of the DM interaction is maximized in materials with large SOC and Ir-O-Ir angle close to 135°. We expect that similar phenomena should be observed in other materials with large SOC and highly non-collinear spin ground states.

METHODS Synthesis
We obtained a high-quality polycrystalline Y 2 Ir 2 O 7 sample using Y 2 O 3 (99.999%; Alfa Aesar) and IrO 2 (99.9%; Sigma Aldrich) powders. As IrO 2 evaporates easily during sintering, we used 4-10% more IrO 2 than suggested by the stoichiometric ratio, Y 2 O 3 : IrO 2 = 1:2. The mixture was pelletized and calcinated in a box furnace at 760°C for 48 h. After regrinding and re-pelletizing, several annealing cycles were performed at temperatures between 900°C and 1030°C, each for 48 h. High-density targets were prepared for optical experiments by pelletizing Y 2 Ir 2 O 7 powder using a high-pressure anvil (NCAP-035; Nanki Engineering) operating at 4.3 GPa. Samples with densities >87% were polished to render the surfaces shiny and flat (see Supplementary Note 6 for details).
X-ray diffraction measurement XRD was used to verify the quality of our Y 2 Ir 2 O 7 crystals, as outlined below. Powder XRD data were obtained using a high-resolution powder diffractometer (D8 Discover; Bruker) operating at wavelengths of 1.54590 and 1.544310 Å (K α1 and K α2 , respectively).

Optical conductivity fitting
We measured the near-normal reflectance spectra R(ω) by Fourier transform infrared spectroscopy (VERTEX 80 v; Bruker) and spectroscopic ellipsometry (V-VASE; J.A. Woollam). To compensate for the effects of rough surface, we normalized R(ω) relative to those measured after in-situ gold evaporation. 66 From R(ω), we calculated the optical conductivity spectra,σ(ω) = σ 1 (ω) + iσ 2 (ω) using the Kramers-Kronig transformation. To track the spectral components, we fitted σ 1 (ω) to a Fano-Lorentz line shape, given by where ω is the angular frequency of the photon, ϵ = (ω − ω 0 )/Γ is a dimensionless parameter for reduced energy, and q is the Fano asymmetry parameter. Here, ω 0 , Γ, and SW = 8 R σ 1 ω 0 ð Þdω 0 are the resonance frequency, linewidth, and spectral weight of the phonon, respectively. For the data measured at 10 and 300 K, the fitted curves are presented as black solid lines in Fig. 2a (for more detailed information, see Supplementary Note 1).

DFT calculations
To obtain the phonon eigenmodes of Y 2 Ir 2 O 7 , we performed DFT+U calculations with the PBEsol exchange correlation function implemented in VASP. 67 We chose the effective U = 1.5 eV to simulate the local Hubbard interaction of Ir, 35 and used the non-collinear DFT formalism to express the AIAO magnetic configuration. We fully optimized the lattice structure and obtained Ir-O-Ir angle θ 0 ≈ 126.9°, as shown in Fig. 4b. We calculated the zone-center phonon modes using the frozen-phonon method. The resulting phonon eigenmodes are summarized in Supplementary Note 3.

Tight-binding model analysis
Using a tight-binding model, we derived J IE (θ) and D(θ), which were required to calculate Δω 0 in equation (22). We derived the effective hopping Hamiltonian, based on the model of Pesin and Balents. 13 However, we applied the effects of strong trigonal distortion of octahedra. We considered two types of electron hopping between the nearest neighbor Ir ions: namely, O-mediated Ir-Ir hopping t oxy and direct Ir-Ir hopping t σ . With t(θ) = a(θ) t oxy + b t σ and v(θ) = a'(θ) t oxy + b' t σ , we could derive the IE and DM interaction: The coefficients a(θ), b, a'(θ), and b' were obtained using the Slater-Koster method 59 (see Supplementary Note 5). We estimated the values of t oxy and t σ by anchoring with the reported values (J IE = 18.8 meV and D = 2.6 meV at θ 0 QCC = 130°) obtained from quantum chemistry calculation (QCC). 60 The calculated J IE (θ) and D(θ) are presented in Fig. 6b.

DATA AVAILABILITY
All relevant data that support the findings of this study are available from the corresponding author on request.