Spin–orbit exciton–induced phonon chirality in a quantum magnet

We performed temperature-dependent Raman measurements to identify the relevant phonons and SOEs. Fig. 2 shows representative Raman spectra taken with the circular polarization combination of ${\sigma }_{-}$ and ${\sigma }_{+}$ for incident and scattered photons. Raman active phonons, including 4 $A_g$ and 5 $E_g$ modes, predicted from group theory are identified in the spectra. The $A_g^{(4)}$ mode is too weak to be observed in the crossed-circular channel but detected in the co-circular channel (included in SI Appendix, Fig. S4). Because of the coupling between different degrees of freedom (i.e., electrons and phonons), the Raman selection rules of phonons in CTO do not follow simple rules determined by the crystal symmetry alone. Below $T_N$, two SOE modes ${\psi }_1$ at 201 cm${}^{-1}$ (25 meV) and ${\psi }_2$ at 195 cm${}^{-1}$ (24 meV) are observed. The splitting between these excitations results from the in-plane exchange mean-field that lifts the degeneracy of the Kramers ground and first excited state. In recent neutron scattering experiments, only the splitting of the Kramers ground state has been observed partially due to limited energy resolution (20, 21). Above $T_N$, SOEs persist and broaden in spectral width. Since the exchange mean-field is modified in the absence of antiferromagnetic order, we use a different notation to represent the SOEs. Both $E_g^{(1)}$(orange) and $E_g^{(2)}$(blue) modes are energetically close to these SOEs and this proximity allows for hybridization between these collective modes. Our model predicts a strong Zeeman energy shift for both $E_g$ phonons, and we expect $E_g^{(1)}$ to exhibit a larger magnetic moment than $E_g^{(2)}$ based on their resonant energy difference.

To quantitatively evaluate spin-lattice coupling, we analyze the temperature-dependent phonon frequencies in the absence of a magnetic field (Fig. 2B). The deviation from the anharmonic model for both $E_g^{(1)}$ and $E_g^{(2)}$ mode is about 1.66 cm${}^{-1}$ and 0.46 cm${}^{-1}$, respectively, and it serves to justify the spin-lattice coupling. In contrast, an adjacent $A_g$ mode at 244 cm${}^{-1}$ exhibits no such frequency deviation from the anharmonic model (data shown in SI Appendix, Fig. S3). Furthermore, the spin-lattice coupling in $E_g$ modes is stronger as they become energetically closer to the SOEs, as shown in Fig. 2A. Above $T_N$, only one SOE mode ψ is observable up to ∼150 K (19). The higher spin-lattice coupling of $E_g^{(1)}$ than that of $E_g^{(2)}$ results from its closer energy to the SOE modes. This interplay of energy scales is also reflected in the phonon magnetic moments as we present below. Chiral phonon modes correspond to the superposition of two components of an $E_g$ mode vibrating in a plane perpendicular to the c-axis. Hence, a chiral $E_g$ phonon can carry angular momentum along the c-axis but not along the a- or b-axis. Our model predicts that the $E_g$ modes will only have a linear magnetic response to an external magnetic field applied along the c-axis. The Raman spectra taken with an external magnetic field applied along different crystal axes shown in Fig. 3 confirm this prediction. Fig. 3A shows helicity-resolved Raman spectra at a magnetic field of 7 T along the c-axis. The phonon energy blue shifts in the ${\sigma }_{-}/{\sigma }_{+}$ channel and red shifts in the opposite ${\sigma }_{+}/{\sigma }_{-}$ channel, as illustrated in the energy diagram (Fig. 3B). In contrast, when the field is applied along either the a- or b-axis, the $E_g^{(2)}$ degeneracy is not lifted (Fig. 3 C and D), validating our model. Furthermore, in Raman spectra measured with an external field along the c-axis but with linearly polarized incident and scattered light, the $E_g^{(2)}$ modes do not split (SI Appendix, Fig. S5). In this case, the superposition between the two chiral phonon modes cancels the energy shift from each mode as the model predicts.

We analyze the angular momentum exchange in the ${\sigma }_{-}/{\sigma }_{+}$ Stokes process associated with an $E_g^{(2)}$ mode as illustrated in Fig. 3B. The modes that appear in the cross-circular channel change the photon angular momentum by 2ħ. A phonon carrying angular momentum ħ is created in the Stokes process. However, the three-fold rotational symmetry of the O${}^{-2}$ ions surrounding the Co${}^{2+}$ ions remedies the conservation of angular momentum (modulo 3ħ): J${}_{\mathit{pt}}/ħ$ + J${}_{\mathit{ph}}$/ħ = 0 (Mod 3) (23–26). The phonons created in ${\sigma }_{-}/{\sigma }_{+}$ and ${\sigma }_{+}/{\sigma }_{-}$ channel carry angular momentum, $J_{}\text{ph}=ħ$ and $J_{}\text{ph}=-ħ$, respectively (27). When an out-of-plane magnetic field is applied, the peaks corresponding to opposite angular momenta shift to different energies, which can be interpreted as the Zeeman splitting of the $E_g$ modes. In the case of an in-plane field, there is no energy difference between the two cross-circular channels, which indicates the absence of a Zeeman effect for an in-plane ${\mu }_0$H field as shown in Fig. 3C.

Finally, we investigate the Zeeman splitting of the $E_g^{(1)}$ and $E_g^{(2)}$ modes in the presence of an external magnetic field along the c-axis at different temperatures. The spectra in Fig. 4A are taken at a magnetic field of 7 T and 12 K, where the Zeeman splitting is $\mathrm{\Delta }{E}_g^{(1)}$ = 7.14 ± 0.04 cm${}^{-1}$ and $\mathrm{\Delta }{E}_g^{(2)}$ = 2.03 ± 0.03 cm${}^{-1}$ for $E_g^{(1)}$ and $E_g^{(2)}$ mode, respectively. Magnetic order and non-trivial magnon topology can also give rise to phonon chirality and phonon-Zeeman splitting (28, 29). If the degeneracy of the $E_g^{(2)}$ mode in CTO is mainly lifted by the antiferromagnetic order, we would expect the phonon splitting to persist in the absence of magnetic field for $T<T_N$. The absence of phonon splitting below $T_N$ in zero magnetic field and a significant Zeeman splitting ($\mathrm{\Delta }{E}_g^{(1)}$ = 4.68 ± 0.07 cm${}^{-1}$ and $\mathrm{\Delta }{E}_g^{(2)}$ = 1.59 ± 0.01 cm${}^{-1}$) observed at 7 T and 50 K (Fig. 4B) suggest the dominant electronic contribution to the magnetic moments of both $E_g^{(1)}$ and $E_g^{(2)}$ modes. The Zeeman splitting of both $E_g$ modes is no longer observable at room temperature where both electronic states in lower Kramers doublet are almost equally populated, and the effect of the magnetic field weakens significantly. This observation further justifies our conclusion that the phonon chirality is of electronic origin.

The splitting of both $E_g$ modes as a function of the magnetic field along the c-axis ($H_c$) follows a simple linear relation, ${\omega }_{\mathit{ph}}^{\pm }={\mathrm{\Omega }}_{+}\pm g_{\mathit{ph}}{\mu }_B{H}_c$. By fitting the slope, we extract $g_{\mathit{ph}}$ = 1.11 ± 0.01 (at 12 K), $g_{\mathit{ph}}$ = 0.68 ± 0.01 (at 50 K), and $g_{\mathit{ph}}$≈ 0 (at 300 K), respectively, for $E_g^{(1)}$. A systematic study of the temperature dependent $g_{\mathit{ph}}$ for both $E_g^{(1)}$ and $E_g^{(2)}$ modes is summarized in Fig. 4E. For both $E_g$ modes, as the temperature is lowered from 300 K, $g_{\mathit{ph}}$ shows a sharp rise around $T_N$ and then saturates at low temperature. This qualitative behavior is captured by our model calculation. We take into account two temperature-dependent factors in the model: population-difference of electronic states and the effective in-plane exchange magnetic field arising from the magnetic order below $T_N$. For $T>T_N$, as T increases, the population difference between two states in the lower Kramers doublet decreases, and thus the magnetic moment of phonons also decreases. For $T<T_N$, an additional effect from the temperature-dependent in-plane exchange magnetic field leads to the saturation of the phonon magnetic moment (SI Appendix). In our model, the $E_g$ splitting is predicted to be directly proportional to spin imbalance resulting from the populated SOEs. While the theory agrees with the observed temperature dependent $g_{\mathit{ph}}$ factor of $E_g^{(2)}$, it fails to reproduce the $g_{\mathit{ph}}$ factor of the $E_g^{(1)}$ quantitatively. Our model omits several features including temperature-dependent lattice distortions and SOE energies, SOE band topology, and more importantly, SOEs’ energies arising from spin-phonon coupling. The hybridization effect is stronger for a phonon mode closer in energy to SOEs. We quantify the degree of hybridization between phonons and SOEs in SI Appendix. Since $E_g^{(2)}$ is further off-resonance with SOEs, these factors contribute less sensitively to its $g_{\mathit{ph}}$ factor.

We summarize the observations of chiral phonons in previous studies. As discussed earlier, two common mechanisms responsible for chiral phonons are i) phonon-induced adiabatic evolution of electronic states and ii) phonon-induced mixing of different electronic energy levels. In the former, phonon-mediated electronic magnetization has both spin and orbital components. The orbital component can be further separated into a non-topological and a topological part. A notable example of recent interest is that of transition metal dichalcogenide monolayers. In these inversion-symmetry broken honeycomb lattices, chiral phonons form at the zone-boundary at the K/K’ valley, and they are topologically trivial (30, 31). Another example is Pb${}_{1-x}$Sn${}_{1-x}$Te thin films, where a large phonon magnetic moment (1 to 3 Bohr magneton) emerges when the electronic bands become topologically non-trivial at $x>0.32$ (6). In the Dirac semimetal Cd₃As₂, a large phonon magnetic moment develops via coupling to a cyclotron resonance over a range of applied magnetic fields when it shifts into resonance with the phonon mode (4). In the case of Cd₃As₂, the second mechanism of electronic hybridization gives rise to the observed phonon chirality.

To find examples more closely related to CTO, we survey other magnetic materials where the strong magnetic response of phonons may derive from coupling with different electronic excitations (1, 6, 7). In rare-earth materials where magnetism originates from f electrons, the CEF causes energy splittings between $4f$ levels that are close to phonon energies, imparting a large magnetic moment to phonons (2, 15). In materials where magnetism derives from d orbitals, CEF splitting is usually of the order of 1 eV, far exceeding the phonon energy. Spin–orbit splitting typically occurs at a lower energy than CEF splitting. In $5d$ systems like the iridates, SOC splitting is still at a much higher energy than phonons. In contrast, spin–orbit splitting in $3d$ systems is typically much smaller than optical phonon energies. Thus, CTO presents an example of $3d$ magnets where the combination of SOC and trigonal distortion leads to strong hybridization between phonons and SOEs.

The underlying mechanism for chiral phonons in CTO bears a strong resemblance to that of the phonon Zeeman effect observed in rare-earth trihalides where the phonon chirality and magnetic moment arise due to the hybridization of phonons with CEF split states (1, 2). In the case of CTO, the involvement of electronic states in this process is primarily influenced by SOC and trigonal distortion. The electronic excitations are comparable in energy to phonons and exchange interactions like “f” electron states in rare-earth systems (19). Moreover, CTO exhibits considerably heightened magnetic transitions, thus, requiring the incorporation of an exchange mean-field approach. This approach, in turn, leads to a more intricate analysis of the phonon Zeeman effect when compared to f electron systems (32). The higher magnetic transition temperature in CTO offers a unique possibility to study the connection between the nature of magnetic order and the phonon magnetic moment as revealed by the $g_{\mathit{ph}}$ temperature dependence (Fig. 4E), unlike in rare-earth paramagnets. This difference can be partially attributed to the presence of an in-plane magnetic anisotropy in CTO. The easy-plane anisotropy leads to an exchange mean-field in the a–b plane that does not lift the degeneracy of chiral phonons but leads to a saturation of phonon magnetic moment at lower temperatures.

The quantum magnet CTO has drawn significant interest recently, especially in the context of topological bosonic excitations. Both magnon and SOE bands are found to be topological in inelastic neutron scattering experiments (19–21). In this work, we unravel another exotic phenomenon in this material, the giant magnetic moments for two phonon modes. We attribute this phonon chirality to the coupling of phonons to SOEs due to their energy proximity. A large magnetic moment allows for enhanced magnetic control of phonons, e.g., selective excitation of certain phonons or manipulation of their transport properties (33). The phonon coupling to a topological SOE we explore here is an exciting topic for further investigations. Theoretical studies suggest that magnon–phonon coupling can change the phonon band topology and affect the chiral phonon transport (34). Similarly, phonons coupled to topological SOEs may also become topological and contribute to the phonon thermal Hall effect. Consistent with this idea, the recent observation of a large thermals Hall signal found in the psuedogap phase of cuprate superconductors was attributed to chiral phonons (35).

This research was primarily supported by the NSF through the Center for Dynamics and Control of Materials: an NSF Materials Research Science and Engineering Center under Cooperative Agreement No. DMR-1720595 and DMR-2308817. Additional support from NSF DMR-2114825 and the Alexander von Humboldt Foundation is gratefully acknowledged by G.A.F. This work was performed in part at the Aspen Center for Physics, which is supported by NSF grant PHY-1607611. G.Y., C.N., and R.H. acknowledge support by NSF Grant No. DMR-2104036. Part of the experiments were performed at the user facility supported by the NSF through the Center for Dynamics and Control of Material under Cooperative Agreement No. DMR-1720595 and The Major Research Instrumentation program DMR-2019130. Work in the Baldini group at UT Austin was primarily supported by the Robert A. Welch Foundation (F-2092-20220331) (to F.Y.G. for data taking) and the United States Army Research Office (W911NF-23-1-0394) (to E.B. for supervision and manuscript writing).