Interlayer magnetophononic coupling in MnBi2Te4

The emergence of magnetism in quantum materials creates a platform to realize spin-based applications in spintronics, magnetic memory, and quantum information science. A key to unlocking new functionalities in these materials is the discovery of tunable coupling between spins and other microscopic degrees of freedom. We present evidence for interlayer magnetophononic coupling in the layered magnetic topological insulator MnBi2Te4. Employing magneto-Raman spectroscopy, we observe anomalies in phonon scattering intensities across magnetic field-driven phase transitions, despite the absence of discernible static structural changes. This behavior is a consequence of a magnetophononic wave-mixing process that allows for the excitation of zone-boundary phonons that are otherwise ‘forbidden’ by momentum conservation. Our microscopic model based on density functional theory calculations reveals that this phenomenon can be attributed to phonons modulating the interlayer exchange coupling. Moreover, signatures of magnetophononic coupling are also observed in the time domain through the ultrafast excitation and detection of coherent phonons across magnetic transitions. In light of the intimate connection between magnetism and topology in MnBi2Te4, the magnetophononic coupling represents an important step towards coherent on-demand manipulation of magnetic topological phases.

Supplementary Note 1: Raman peak assignment and eigenvectors 1 We start with a systematic analysis of Raman phonon spectra, shown in Supplementary Figure 1a. The non-2 magnetic unit cell contains seven atoms, and thus there are 21 phonon modes in total, consisting of 18 3 optical and 3 acoustic modes. Using representation theory, these can be decomposed into irreps of the point 4 group 3 � m. Of these, only the and 1 modes are Raman active. Polarized Raman spectroscopy 5 measurements are used to readily identify these modes based on their different selection rules. In particular, 6 modes have non-vanishing diagonal Raman tensor components and are thus visible under both parallel-7 and cross-polarized configurations, whereas the 1 modes have only diagonal Raman tensor components 8 and are visible only under the parallel-polarized configuration. We did not observe any dependence on the 9 in-plane crystallographic orientation.

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The phonon eigendisplacements at the Γ point, calculated using density functional theory simulations, show 1 that 1 phonons have purely out-of-plane ionic motions, whereas phonons have purely in-plane ionic 2 motions. 3 Supplementary Note 2: Anomalous temperature-dependence of ( ) mode 1 In Supplementary Figure 2a, we plot the Raman spectra measured at 15 K and 300 K, normalized to the 2 height of the (3) peak at ~113 cm -1 , for convenience. We note that the result identified below is 3 independent of the choice of normalization. In general, phonon peaks in Raman spectra broaden with 4 increasing temperature due to increased phonon-phonon scattering, with resultant lower peak heights. This 5 is visible for instance in the 1 (3) peak at ~145 cm -1 . On the other hand, the scattering intensity of the 1 6 mode exhibits an anomalous temperature-dependence, with a dramatic decrease in height and integrated 7 intensity, with decreasing temperature (see Supplementary Figure 2b). It is apparent that this decrease in 8 amplitude is independent of the choice of normalization. The amplitude does not show any clear correlation 9 with the magnetic transition at T N = 24 K.

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The amplitude of the 1 (1) peak, fit to a Fano lineshape, as outlined in the Methods section. The scattering intensity associated with the zone-boundary 1 (1) mode is clearly visible in the field-20 dependent Raman spectra in Fig. 1d in the main text. In the temperature-dependent Raman spectra in Fig.  21 1c however, the anomalous temperature-dependence, described above, appears to swamp the small zone-22 boundary scattering intensity. 23 Supplementary Note 3: Field-dependence of ( ) , ( ) , and ( ) spectral weights 1 The integrated intensities of the respectively, as a function of magnetic field. The integrated intensities were obtained by fitting individual 3 spectra following the procedure outlined in the Methods section, with the error bars denoting the standard 4 deviation in fit values. We note a small dip in the (2) intensity at the spin-flop critical field of 3.7 T. 5 Outside of this, the three modes shown here exhibit no clear field-dependent behavior above the 6 experimental and fitting uncertainty. In particular, there is no signature of coupling to the antiferromagnetic 7 order parameter and the associated zone-boundary phonons. We write down minimal lattice and spin Hamiltonians 2 to describe a generalized magnetophononic 3 coupling. Consider the lattice Hamiltonian described by the harmonic approximation, 4 where is the displacement along the phonon normal mode , is the reduced mass, is the frequency, 6 and is the number of unit cells. The magnetic ground state energy described by a Heisenberg-like 7 Hamiltonian, 8 where and are spin site indices, and is the isotropic exchange interaction between spins at and . 10 When this is perturbed by a zone-center optical phonon , the perturbed magnetic energy can be derived 11 by considering the derivatives of with respect to the phonon normal mode displacement . Expanding 12 upto second order in , the perturbed exchange interaction is 13 Here, the first order term, in the specific case of = ⊥ is responsible for the magnetophononic wave-15 mixing described in detail in the main text (Supplementary Eq. 1 and Eq. 2). 16 The second order term, proportional to 2 , renormalizes the harmonic term in the lattice energy, resulting 17 in spin-induced phonon frequency changes. Separating the in-plane and out-of-plane exchange couplings, 18 denoted by and ⊥ , respectively, where = 1 ,2, … are the first-and second-nearest-neighbors and so 19 on, and assuming small spin-induced energy shifts i. e. + 0 ≈ 2 0 , the renormalized phonon 20 frequency is given by 21 The above expression shows the renormalization of the phonon frequency due to spin order along different 23 directions, through the respective exchange couplings. Under a mean-field approximation, Supplementary 24 Eq. 4 simplifies to ν α − 0 ∝ ⟨ 2 ⟩. 25 Experimentally, we observe such a spin-induced phonon frequency renormalization in the 1 (1) mode. The 26 phonon frequencies are first extracted as a function of temperature, using the fitting procedure outlined in 27 the Methods section of the main text. We then account for phonon-phonon interactions by fitting the 28 temperature-dependent phonon frequencies to that of a (cubic) anharmonic phonon, given by ( is the phonon frequency renormalized by anharmonic (phonon-phonon) 30 interactions, is the temperature, 0 is the bare phonon frequency, and is a mode-specific fitting 31 constant. Interestingly, we note that a previous study 3 on atomically thin flakes of MnBi 2 Te 4 reported a negative spin-11 induced frequency renormalization of the 1 (1) mode, contrary to the positive frequency renormalization 12 observed in the bulk crystals used in our study. This difference may possibly be due to changes in the 13 electronic and magnetic structure as a function of sample thickness in the 2D limit. 14 The strong magnetophononic coupling observed in the 1 (2) mode in our magneto-Raman measurements 15 suggests that it too might exhibit a significant spin-induced frequency shift. Unfortunately, the spectral 16 overlap between the 1 (2) and (3) modes (see Supplementary Figure 1a) and strong 1 (2) zone-boundary 17 scattering intensity below T N hinders a similar temperature-dependent frequency analysis for the 1 (2) mode.

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None of the other observed Raman phonons exhibit a spin-induced frequency renormalization above the 19 experimental uncertainty. 20

Supplementary Note 5: Resonant Raman effects
1 Resonant Raman effects may potentially give rise to temperature-and field-dependent artifacts in phonon 2 peak intensities due to changes in the electronic band structure across phase transitions. In order to rule out 3 such an explanation for the phenomena reported in Fig. 1 and Fig. 2, we investigate resonant Raman effects 4 in MnBi 2 Te 4 by measuring phonon spectra at different laser excitation energies. In Supplementary Figure  5 5

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It is observed that phonon peak intensities indeed change as a function of the laser excitation energy, 14 however, these changes occur across all the observed phonon modes, i. e. the three 1 modes as well as 15 the three modes. It is clear that this result is independent of the choice of normalization. In contrast, the 16 temperature-and field-dependent magnetophononic effects observed in our study are only in the 1 (1) and 17 1 (2) modes, with negligible changes in the scattering intensities of other modes. Our observations reported 18 in Fig. 1 and Fig. 2 are thus inconsistent with resonant Raman effects. 19 Furthermore, upon tracking the 1 (2) mode across the PM → AFM transition at 24 K using the 1.58 eV (785 20 nm) laser excitation, it is found that the Raman scattering intensity exhibits quantitatively the same behavior 21 (see Supplementary Figure 6) as with the 1.96 eV (633 nm) laser excitation (see Fig. 1c) -i. e. the 1 (2) 22 scattering intensity is enhanced by around 35% in the AFM phase. This is additional evidence that the 23 observed phenomenon is inconsistent with resonant Raman effects, wherein different excitation energies 24 would give rise to qualitatively different temperature-dependent intensity changes. It is instead consistent 25 with an effect arising from the AFM order, as in our model of magnetophononic wavemixing. 26 It is useful to consider the exchange energies involved in various magnetic phase transitions in MnBi 2 Te 4 . 27 The dominant in-plane nearest neighbor exchange coupling is 0.12 meV, whereas the interplanar exchange 28 coupling is an order-of-magnitude weaker 4 . The temperature-driven PM → AFM transition is accompanied 29 by significant magnetic energy changes due to the in-plane ordering of spins, and the large in-plane 30 exchange coupling. On the other hand, the in-plane ordering remains unchanged in the out-of-plane 31 magnetic field-driven AFM → FM transition, with only the interplanar magnetic order being modulated. 32 The accompanying magnetic energy changes are thus an order of magnitude weaker than in the PM → AFM 1 transition. Hence it is expected that the associated electronic structure changes as a function of out-of-plane 2 magnetic field would also be correspondingly small, minimizing artifacts due to resonant Raman effects. 3 This assertion is validated in our work, where we find that the scattering intensities of the modes and 4 the 1 (3) mode are unchanged as a function of out-of-plane magnetic field within the experimental 5 uncertainty, as outlined in Section S3, allowing us to identify magnetophononic zone-folding in the 1 (1) 6 and 1 (2) peaks. Importantly, the phenomena observed in Fig. 1 and Fig. 2 are correlated not with magnetic 7 order itself, but specifically with AFM order. The zone-boundary phonon intensity vanishes in the FM 8 phase. In fact, as the results in Fig. 2 show, the zone-boundary intensity of the 1 (2) phonon quantitatively 9 tracks the AFM order in both the temperature-and magnetic field-dependent experiments. 14 Based on the above arguments, in order to rule out resonant Raman effects and isolate peak intensity 15 changes due to magnetophononic coupling, it is essential to measure and analyze phonon scattering 16 intensities as a function of both temperature and magnetic field, as carried out in the present work. In order to eliminate the possibility that the anomalous scattering intensities observed in our work (plotted 3 in Fig. 1 and Fig. 2) are due to magnons, we carry out a polarization analysis. 4 Magnons, by virtue of breaking time-reversal symmetry necessarily have off-diagonal terms in the Raman 5 tensor 5 . In MnBi 2 Te 4 , this is associated with modes, as opposed to 1 modes which are fully symmetric 6 and have only diagonal components. The symmetry associated with Raman scattering intensity can be 7 identified as 1 or using polarized Raman measurements, as in Supplementary Note 1. Here, we focus 8 on the 1 (2) mode. In Supplementary Figure 7, we show Raman spectra obtained below and above the AFM 9 ordering temperature T N = 24 K, corresponding to parallel-polarization, which is sensitive to both 1 and 10 modes, and cross-polarization, which is sensitive only to modes. Our results clearly show that the 11 anomalous scattering intensity overlapped with the 1 (2) phonon in the AFM phase has an 1 symmetry, 12 since it is absent in the cross-polarized channel. This rules out the possibility that it is due to a magnon. It 13 is instead consistent with our interpretation in terms of scattering intensity due to 1 (2) zone-boundary 14 phonons.

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The above inference is also consistent with magnon dispersions measured using inelastic neutron 20 scattering 4 . The dispersion shows that zone-center magnons are at around 1 meV (~8 cm -1 ), whereas the 21 highest energy zone-boundary magnons are at 3 meV (~25 cm -1 ). The low energy of zone-center magnons 22 rules out the possibility of one-magnon resonance interfering with phonon peaks. A two-magnon resonance 23 may plausibly interfere with the 1 (1) mode at 47 cm -1 but would be too low in energy to affect the 1 (2) 1 mode at 115 cm -1 , ruling it out as an explanation for the observed phenomena. Two-magnons are also 2 typically associated with a broad continuum of excitations rather than a well-defined peak, a feature that 3 we do not observe in our experiments. 4

Supplementary Note 7: Pump-probe measurements -Fluence-dependence 1
The pump-probe measurements outlined in the main text show phonon excitation via a displacive 2 mechanism, where the ultrafast excitation of carriers by the pump pulse shifts the quasiequilibrium ionic 3 coordinates, generating coherent phonons. Here, the amplitude of coherent phonons is directly proportional 4 to the pump-induced carrier density, i. e. the absorbed fluence. In this context, field-dependent optical 5 conductivity changes may influence coherent phonon amplitudes, in addition to the magnetophononic 6 coupling highlighted in the main text. We account for such magnetic-field dependent changes in absorbed 7 fluence by normalizing the pump-probe measurements with respect to the pump-induced carrier density. 8 The carrier density can be tracked by the maximum amplitude of the transient reflectivity trace, which 9 occurs at a time delay of ~0.9 ps. In Fig. 4 of the main text, all the pump-probe traces are normalized with 10 respect to this amplitude. Finally, we note that outside of magnetic-field dependent changes in absorbed fluence, there may 5 potentially be additional magnetodielectric effects that change the electron-phonon interactions and thus 6 the Raman susceptibility, which can affect coherent phonon generation. To lowest order, such changes may 7 be phenomenologically described by a magnetodielectric effect of the form the electrical susceptibility, is magnetodielectric coefficient, and is the net magnetization. Below, we 9 explore possible changes to electron-phonon interaction due to such a magnetodielectric effect. In the 10 interest of conceptual clarity, we consider a simple -dependence = , where is the magnetic 11 susceptibility and is the external magnetic field, the Raman susceptibility of a phonon (which determines 12 the coherent phonon amplitude via a Raman-like displacive excitation) can then be written as = amplitude would then be a form of indirect magnetophononic coupling. Based on our current pump-probe 17 experimental data, we cannot completely rule out that such indirect magnetophononic effects also have a 18 contribution, in addition to the direct magnetophononic effects highlighted in our manuscript.

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In general, the intensity of a Bragg reflection in an electron diffraction experiment is given by 12 where the summation is over atoms in the unit cell (indexed by ), is the scattering wavevector, is the 14 atomic structure factor for electron diffraction for atom , is the isotropic Debye-Waller factor for atom 15 , and is the position of atom in the unit cell. The Debye-Waller factor is given by = ⟨ 2 ⟩, which is 16 the root-mean-square displacement of atom about its mean position. A representative static diffraction 17 pattern from a ~100 nm flake of MnBi 2 Te 4 oriented along the (0 0 1) crystallographic direction is shown in 18 Supplementary Figure 9a. In the main text, we use the transient Debye-Waller time constant from our UED measurements to establish 16 the timescale of lattice thermalization. However, the optical pump-probe measurements reported in the 17 main text use a much lower fluence, of 0.1 mJ/cm 2 , as opposed to 7 mJ/cm 2 used in the UED measurements 18 reported above. In this context, we report the thermalization time constants from our UED measurements 19 as a function of fluence, in Supplementary Figure 9d. The time constants are largely unchanged from 5 to 20 9 mJ/cm 2 , with a slight increase at lower fluences. 21 Such a behavior is consistent with increased phonon-phonon scattering at higher fluences 6 . It is expected 22 then that the thermalization time constant at the low fluences used in our optical measurements, with their 23 correspondingly lower phonon populations, would likely be even higher than that extracted from the UED 24 measurements; i. e. the UED time constant sets a lower bound for the phonon thermalization time. This 25 supports our assertion that phonon subsystem remains in a nonequilibrium state through the entire time 26 delay range measured in our study. 27 28