Abstract
The emergence of magnetism in quantum materials creates a platform to realize spinbased 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 MnBi_{2}Te_{4}. Employing magnetoRaman spectroscopy, we observe anomalies in phonon scattering intensities across magnetic fielddriven phase transitions, despite the absence of discernible static structural changes. This behavior is a consequence of a magnetophononic wavemixing process that allows for the excitation of zoneboundary 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 MnBi_{2}Te_{4}, the magnetophononic coupling represents an important step towards coherent ondemand manipulation of magnetic topological phases.
Introduction
The realization of magnetic order in functional quantum materials creates a rich platform for the exploration of fundamental spinbased phenomena, as exemplified in strongly correlated materials^{1}, multiferroics^{2}, and more recently, magnetic topological materials^{3}. As such, these materials hold great promise for application in spintronics, magnetic memory, and quantum information technology. A new paradigm has recently emerged with the discovery of atomically thin magnets, derived from layered, quasitwodimensional materials^{4}. In such materials, magnetic order is characterized by strongly anisotropic exchange interactions, with interlayer exchange coupling that is an orderofmagnitude weaker than the inplane exchange coupling. The weak interlayer exchange coupling offers a high degree of tunability in the twodimensional limit, enabling the realization of phenomena such as magnetic switching via electric fields^{5} and electrostatic doping^{6}. Such tunability could potentially be made even more potent in combination with additional functionalities such as those outlined above. For instance, the Mn(Bi,Sb)_{2n}Te_{3n + 1} family of layered antiferromagnets is the first experimental realization of intrinsic magnetic order in topological insulators^{7,8,9}. The interlayer magnetic order is intimately connected to the band topology, with experimental demonstration of switching between quantum anomalous Hall and axion insulator states^{10}, and realization of a fielddriven Weyl semimetal state^{11}. In this context, the discovery of new, efficient coupling pathways between the interlayer exchange and other microscopic degrees of freedom would not only add to the rich spectrum of lowdimensional magnetic phenomena, but also potentially unlock pathways for the dynamic manipulation of magnetism and band topology.
In this work, we observe that interlayer magnetic order in MnBi_{2}Te_{4} is strongly coupled to phonons, manifesting in the optical excitation of zoneboundary phonons that are otherwise forbidden due to the conservation of momentum. This magnetophononic response is a consequence of a coherent wavemixing process between the antiferromagnetic order and A_{1g} optical phonons, as determined from equilibrium and timedomain spectroscopy across temperature and magnetic fielddriven phase transitions. Our microscopic model based on firstprinciples calculations reveals that this phenomenon can be attributed to phonons modulating the interlayer exchange coupling.
Results
Spectroscopic evidence of magnetophononic coupling
MnBi_{2}Te_{4} exhibits magnetic order below a temperature of T_{N} = 24 K, with inplane ferromagnetic coupling, and outofplane antiferromagnetic (AFM) coupling^{12}, as shown in Fig. 1a. With an applied outofplane magnetic field, a spinflop transition occurs at 3.7 T, developing into a fully polarized ferromagneticlike state (FM) at a critical field of 7.7 T^{12}. We first present measurements of the phonon spectra across the magnetic phase transitions in MnBi_{2}Te_{4}, using magnetoRaman spectroscopy. The full polarized Raman phonon spectra, selection rules, and peak assignments can be found in Supplementary Note 1. Our peak assignment is fully consistent with a previous study^{13} that investigated Raman phonons in thin flakes of MnBi_{2}Te_{4} as a function of number of layers. Here we focus on two fully symmetric “A_{1g}” phonon modes at frequencies of 49 and 113 cm^{−1}, labeled A_{1g}^{(1)} and A_{1g}^{(2)} respectively. The phonon eigendisplacements, calculated using density functional theory (DFT) simulations, are shown in Fig. 1b. Representative spectra at 0 T, in the AFM phase at 15 K and the paramagnetic (PM) phase at 35 K, are shown in Fig. 1c, d, respectively. We observe that the A_{1g}^{(2)} mode clearly exhibits an anomalous increase in scattering intensity in the AFM phase, which has not been reported in previous studies^{13}. The temperaturedependence of the A_{1g}^{(1)} mode is discussed in detail in Supplementary Note 2. In the following, we focus on the magnetic fielddependent behavior. At a magnetic field of 9 T, where MnBi_{2}Te_{4} is in the fully polarized ferromagnetic (FM) state, the spectral weight of both modes decreases, as shown in Fig. 1e, f. This is highlighted by subtracting the spectrum at 9 T from the spectrum at 0 T and plotting the residual in Fig. 1g, h. In Fig. 1i, j, the residual is plotted as a function of magnetic field H, upon subtracting the 9 T spectrum. A clear correlation is observed between the residual scattering intensity of the A_{1g} modes and the critical magnetic fields for the spinflop and FM transitions, denoted by dashed white lines.
The fractional change in integrated intensity of the A_{1g}^{(2)} mode is plotted as a function of temperature in Fig. 2a (green dots). The integrated intensity follows the AFM order parameter, tracked by the (1 0 5/2) neutron diffraction Bragg peak^{14} (purple dots). The gray line is a fit to the power law \(I\,\propto\, {\left(1\,\,\frac{T}{{T}_{N}}\right)}^{2\beta }\), with β = 0.35 as in the reference^{14}. Furthermore, a plot of the scattering intensity of the A_{1g}^{(1)} and A_{1g}^{(2)} modes (Fig. 2b) as a function of magnetic field reveals the fractional change in integrated intensities of both modes tracks the AFM order parameter^{15} across the spinflop transition at 3.7 T, and into the fully polarized ferromagnetic state above 7.7 T. The integrated intensities of the A_{1g}^{(1)} and A_{1g}^{(2)} modes increase by fractions of 0.15 and 0.3 respectively, in the AFM phase, as compared to the FM phase at 9 T. Additionally, the fractional increase in the A_{1g}^{(2)} intensity as estimated from the PM to AFM transition and FM to AFM transition in Fig. 2a, b, respectively, are of the same magnitude, pointing to a common origin. Importantly, within the limits of our experimental uncertainty (error bars in plots), we do not observe such large changes in the integrated intensity on any of the other Raman phonons (see Supplementary Note 3 for detailed fielddependent data). Below, we show that the experimentally observed temperature and fielddependent evolution of scattering intensity is consistent with the excitation of ‘forbidden’ zoneboundary modes of the A_{1g}^{(1)} and A_{1g}^{(2)} phonon branches.
The AFM order along the outofplane direction (crystallographic caxis) results in a magnetic unit cell that is double the size of the crystallographic unit cell, as shown in Fig. 3a. In contrast, in the highfield FM state (and the paramagnetic state), the magnetic unit cell is identical with the crystallographic unit cell, as in the paramagnetic state. This behavior manifests in the anomalous fielddependent scattering intensity of the A_{1g} modes, which follows the AFM order parameter with the magnetic unit cell doubling resulting in a folding of the phonon Brillouin zone, allowing for the optical detection of zoneboundary phonon modes. DFT simulations of the phonon dispersion along the outofplane direction reveal a flat dispersion for the A_{1g}^{(2)} mode, and a small dispersion for the A_{1g}^{(1)}, consistent with the weak interlayer van der Waals interaction, and our experimental results, denoted in Fig. 3b using bold circles. This supports our assignment of the anomalous scattering intensity as zoneboundary modes. We also consider and rule out alternative explanations for the observed temperature and magnetic fielddependent scattering intensity changes, such as resonant Raman effects (see Supplementary Note 5) and possible magnon resonances overlapping with the considered phonons (see Supplementary Note 6).
Magnetic unit cell doubling resulting in the activation of zoneboundary phonons is unexpected given the absence of a structural phase transition. Refinement based on neutron diffraction at 10 and 100 K shows no structural unit cell doubling across the AFM transition, and no changes to the unit cell coordinates to within 10^{−3} of the lattice parameters^{14}. The negligible change in the spectra of other Raman phonons in MnBi_{2}Te_{4} is also consistent with the absence of a structural transition of any kind, and points instead to a mechanism that is modedependent.
Microscopic model of magnetophononic wavemixing
In general, zoneboundary modes are optically inactive or “forbidden” due to the conservation of crystal momentum. Photons in the visible part of the spectrum have negligible momentum in comparison with the crystal Brillouin zone, and thus momentum conservation dictates that only zero momentum (i.e., zonecenter) excitations can be generated and detected in firstorder scattering processes. This is shown schematically for Raman scattering in Fig. 3c. This selection rule can be overcome in the presence of other finitemomentum waves in the crystal, as observed for instance in the case of structural distortions that double the crystallographic unit cell^{16,17,18}. However, as noted above, MnBi_{2}Te_{4} does not exhibit any structural transition. Instead, we propose that the crystal momentum is provided by the AFM order, via a magnetophononic wavemixing process. This is shown schematically in Fig. 3c, where the AFM crystal momentum q_{AFM} = 2π/2c interacts with the phonon crystal momentum, allowing for the excitation of zoneboundary (q = π/c) phonons.
Magnetophononic wavemixing requires a sufficiently strong scattering crosssection to be observable. This scattering crosssection can typically be written in terms of an interaction term in the free energy. For example, the Raman scattering process is due to a coupling of the incident \(({E}^{i})\) and reflected \(({E}^{r})\) electric fields to a distortion u along a phonon normal mode, via the susceptibility \({\chi }_{e}\) (i.e., \(F\,=\,\left(\frac{d{\chi }_{e}}{{du}}u\right){E}^{i}{E}^{r}\)). In the case of a finitemomentum structural distortion, phonons couple to the structural distortion through elastic interactions. Analogously, in our model of magnetophononic wavemixing, phonons couple to the AFM order by modulating the interlayer exchange interaction \({J}^{\perp }\). The corresponding interaction term in the free energy can be obtained by first writing down a Heisenberglike Hamiltonian for the spin energy, \(H\,=\,\mathop{\sum}\limits_{{ij}}{J}_{{ij}}{S}_{i}.{S}_{j}\), where \({J}_{{ij}}\) is the exchange coupling between spins at sites i and j. Since the coupling is to an outofplane antiferromagnetic spin wave, we focus on the interlayer (outofplane) exchange coupling \({J}^{\perp }\) (only nearestneighbor interlayer interactions are considered). If a phonon modulates the interlayer exchange interaction, the perturbed exchange coupling \({J}^{\perp\prime}\) can be written as
Equation 1 is a special case of what is broadly referred to in the literature as “spin–phonon coupling” (see Supplementary Note 4 for the interpretation of higherorder terms in terms of phonon frequency renormalization). Based on this, the free energy term that couples the antiferromagnetic spin wave to the phonon is, to first order,
where i and i + 1 correspond to nearestneighbor spin pairs in the outofplane direction. It is clear that the magnitude of this coupling directly depends on \(\frac{d{J}^{\perp }}{{du}}\). In other words, a magnetophononic wavemixing is possible only when the phonon mode under consideration sufficiently modulates the interlayer exchange coupling.
A microscopic basis for this model can be obtained using DFT simulations. We simulate the modulation of the interlayer exchange coupling \({J}^{\perp }\) by the six Raman phonons of MnBi_{2}Te_{4}, which include three A_{1g} modes (pure outofplane eigendisplacements), and three E_{g} modes (pure inplane eigendisplacements, see Supplementary Fig. 1b for eigendisplacements). The results, shown in Fig. 3d, indicate a striking dichotomy between the outofplane A_{1g} modes and inplane E_{g} modes. The A_{1g} modes exhibit an orderofmagnitude larger modulation of \({J}^{\perp }\) than the E_{g} modes. Furthermore, the A_{1g}^{(2)} mode has by far the largest influence on \({J}^{\perp }\), consistent with our experimental observation of zoneboundary scattering intensity. A quantitative comparison of this model with our experimental results is possible. This is accomplished by defining an experimental magnetophononic scattering crosssection σ, as the ratio of the integrated intensity of the zoneboundary mode (i. e. the residual spectra in Fig. 1g, h) to that of the zonecenter mode (spectra at 9 T in Fig. 1e, f). The scattering crosssection is compared to the calculated interaction term, \(\left\frac{d{J}^{\perp }}{{du}}\right\). The plotted results in Fig. 3e show a good agreement between theory and the experiment. In particular, the model reproduces the experimental observation of the A_{1g}^{(2)} mode exhibiting the largest zoneboundary scattering intensity. We note that no signature of a zoneboundary mode was observed in the A_{1g}^{(3)} branch within the experimental uncertainty (see Supplementary Note 3). Finally, also in agreement with the theoretical prediction, no E_{g} zoneboundary modes were experimentally observed, i.e., σ = 0 for all E_{g} modes, within the experimental uncertainty (see Supplementary Note 3).
The theoretical results outlined above can be rationalized in terms of microscopic interlayer exchange pathways. In general, the exchange coupling across a van der Waals (vdW) gap is understood to be the result of a process named “supersuperexchange” (SSE)^{19}. In SSE, given that the interlayer exchange interaction is usually much weaker than the intralayer exchange interaction, the two can be effectively decoupled. The individual quasitwodimensional layers are treated as macroscopic magnetic moments established by the intralayer superexchange (shown in pink in Fig. 3f), which couple across the vdW gap via the weaker interlayer exchange (shown in blue in Fig. 3f). As in any exchange process, geometrical parameters that influence the relevant hopping integrals play a major role. In superexchange, the angle between magnetic ions and its ligands mediates the superexchange, in this case the Mn–Te–Mn bond angle θ shown in Fig. 3f. These structural superexchange interactions are further controlled by orbital hybridization with cationic Bi p states tuned by the nearestneighbor ions across the vdW gap^{20}, in this case, determined by the Te–Te distance Δ shown in Fig. 3f, to stabilize the FM interlayer coupling in MnBi_{2}Te_{4}.
We first note that A_{1g} modes in MnBi_{2}Te_{4} modulate Δ, whereas E_{g} modes do not, an observation that accounts for the dichotomy of their respective influence on \({J}^{\perp }\). Of the A_{1g} modes, examining the eigenvectors in Fig. 1b and Supplementary Fig. 1b, A1g^{(2)} exhibits the largest modulation of the Mn–Te–Mn bond angle θ. The modulation of θ by the A_{1g}^{(2)} mode is a factor of 2 larger than by A_{1g}^{(1)}, which in turn is a factor of 5 larger than by A_{1g}^{(3)}. This rationalizes the trend seen in the calculated \(\frac{d{J}^{\perp }}{{du}}\) in terms of the SSE pathways.
Timedomain signatures of magnetophononic coupling
Finally, we investigate magnetophononic coupling by direct measurement of phonons in the time domain. To do this, we carry out “pumpprobe” experiments to generate and detect coherent optical phonons as a function of magnetic field (see schematic in Fig. 4a). Excitation with ultrafast optical pump pulses (1.55 eV, 50 fs) results in the generation of coherent phonon oscillations. A second, timedelayed probe pulse (1.2 eV, 50 fs) measures pumpinduced changes in the transient reflectivity (ΔR/R). The transient reflectivity is sensitive to changes in carrier density and coherent phonons. These measurements are carried out at 2 K, as a function of magnetic field from 0 to 6.4 T, across the spinflop transition. The transient reflectivity, shown in Fig. 4b, exhibits an initial subpicosecond dip, followed by a slow relaxation. Overlayed on this are multiple distinct coherent oscillation components that (as described below), correspond to the A_{1g}^{(1)} and A_{1g}^{(2)} phonons. We normalize the pumpprobe reflectivity traces with respect to their maximum amplitudes, to account for fielddependent variation in the absorbed fluence, and thus photocarrier density, which can influence coherent phonon amplitudes (see Supplementary Note 7 and Supplementary Fig. 8 for detailed discussion). Upon subtracting biexponential fits (black line fit to 0 T data in Fig. 4b shown as a representative example), we observe that the normalized phonon oscillation amplitudes in the residual ΔR/R in Fig. 4c visibly decrease with increasing magnetic field, much like the phonon spectral weights measured using Raman spectroscopy.
The individual oscillatory components are obtained by fitting the residual ΔR/R to the sum of two exponentially decaying sinusoidal functions (see Methods) as shown for the representative 0 T data in Fig. 4d The individual sinusoidal functions, shown in Fig. 4e, are readily identified as the A_{1g}^{(1)} and A_{1g}^{(2)} modes at 1.47 THz (49 cm^{−1}) and 3.44 THz (115 cm^{−1}), respectively. Plotting the amplitudes of the two coherent phonon modes as a function of magnetic field in Fig. 4e, it is clear that both modes track the AFM order parameter denoted by the solid gray line, in striking similarity to the fielddependent change in the Raman scattering intensities.
The detection of coherent phonons in pumpprobe experiments occurs through a process that is identical to spontaneous Raman scattering^{21,22}. The generation of coherent phonons can also be described within a Raman formalism, with the real and imaginary parts of the Raman tensor responsible for phonon excitation in transparent and absorbing materials, respectively^{21}. The similarity of the magneticfielddependent coherent phonon amplitudes in Fig. 4f to the static Raman scattering intensities in Fig. 2b thus suggests that these are a consequence of the same mechanism, namely the excitation of zoneboundary phonons via the crystal momentum associated with the antiferromagnetic order.
For resonant excitation of MnBi_{2}Te_{4} with 1.55 eV pulses, phonon excitation through the imaginary part of the Raman tensor may be physically thought of in terms of a “displacive” excitation^{23}, where the ultrafast excitation of carriers by the pump pulse shifts the quasiequilibrium coordinates of the lattice in a spatially and temporally coherent manner, generating coherent phonons. Within this picture, magnetophononic zonefolding as described in the previous section would allow for the generation of both zonecenter as well as nominally zoneboundary A_{1g} modes. Additionally, the electronic excitation that shifts the quasiequilibrium coordinates may itself have a q_{z} = π/c component owing to the contrast in spinsplit electronic bands in alternating layers, acting as a direct driving force for the generation of zoneboundary phonons. Unfortunately, the small frequency splitting of the A_{1g} modes precludes the explicit resolution of zoneboundary phonons in the time domain. Nonetheless, it is clear from Fig. 4f that the coherent phonons track the AFM order parameter in accord with the magnetophononic wavemixing proposed here.
We note that in general, phonons in timedomain measurements are expected to exhibit qualitative deviations from steadystate spectroscopy, owing to the nonequilibrium nature of the former. While the ultrafast carrier excitation in displacive phonon excitation is itself a manifestly nonequilibrium process, additional deviations may emerge from nonequilibrium phonon interactions. We directly measure the timescale of phonon equilibration using ultrafast electron diffraction (see Methods). Here, pumpinduced changes in the rootmeansquare displacements 〈u^{2}〉 of ions through carrierlattice and lattice thermalization appear in the transient intensity of Bragg peaks through the DebyeWaller effect (see Supplementary Note 8). These measurements require an orderofmagnitude higher pump excitation fluence than the optical pumpprobe measurements (see Methods) in order to produce a discernible signal. Regardless, these high fluence measurements set a lower bound for the phonon thermalization time, as discussed in Supplementary Note 8. As a representative sample, we show in Fig. 4g, the transient intensity of the (2 2 0) Bragg peak, with the evolution of the peak intensity fit to an exponential decay (black line). The results indicate that phonon populations indeed remain in a nonequilibrium state through the entire time delay range considered. It is noteworthy that clear signatures of magnetophononic coupling are observed even under such nonequilibrium conditions. Finally, we mention that there may possibly be additional contributions to the coherent phonon amplitudes from magnetodielectric effects which are not explicitly accounted for here. We discuss the possible contributions to coherent phonon amplitudes due to such an effect Supplementary Note 7.
Discussion
We have demonstrated that optically “forbidden” zoneboundary phonons are observed due to magnetophononic wavemixing in MnBi_{2}Te_{4}. While it is uncommon for purely magnetic unit cell doubling to give rise to phonon zonefolding effects, such signatures were first observed in transition metal dihalides^{24}. These observations were rationalized in terms of phenomenological models of electronphonon coupling that took into consideration phonon modulation of the spin–orbit coupling and exchange interactions^{24,25}. Our model instead considers the scattering crosssection between the AFM order and phonons, arriving at qualitatively similar conclusions. Importantly, our work provides a description of such a model using firstprinciples theory. The excellent agreement between the theory and experimental results not only validates the model, but also provides a microscopic basis for the observed phenomena in terms of SSE interlayer exchange pathways. Our work may also help rationalize similar phenomena recently reported^{26,27} in other quasitwodimensional magnets such as CrI_{3} and FePS_{3}.
Our discovery is especially of significance in light of the critical role played by tunable interlayer exchange interactions in layered magnetic materials. For instance, in MnBi_{2}Te_{4}, the interlayer magnetic ordering can drive topological phase transitions between quantum anomalous Hall and axion insulator states. Our work unlocks the possibility of controlling the interlayer magnetic ordering in MnBi_{2}Te_{4} by exploiting the strong coupling of A_{1g} phonons to \({J}^{\perp }\). A promising route towards the ultrafast control of magnetism in MnBi_{2}Te_{4} is the use of resonant THz excitation to drive large amplitude distortions along A_{1g} modes, as opposed to employing carrierbased mechanisms (such as displacive excitation) that suffer from ultrafast heating effects, which limit the amplitude of coherent phonons. This may be through anharmonic coupling to Ramanactive modes^{28}, or alternatively through sumfrequency ionic Raman scattering^{29}. Such mechanisms based on resonant coupling have been used to drive ultrafast lightinduced magnetic oscillations and phase transitions, as experimentally demonstrated in other materials^{30,31,32,33,34,35,36}. Experimental studies^{37} on Bi_{2}Se_{3}, a material closely related to MnBi_{2}Te_{4}, have demonstrated the feasibility of ionic Raman scattering as a way to drive large amplitude oscillations along Ramanactive modes. Recent theoretical work^{38} has outlined an approach based on anharmonic phonon interactions in MnBi_{2}Te_{4}. In particular, it was shown that resonant excitation of an IRactive A_{2u} phonon (at a frequency of 156 cm^{−1} = 4.7 THz) could drive large amplitude oscillations, which via anharmonic coupling, would drive a unidirectional distortion along Ramanactive A_{1g} modes such as the ones identified in the present work. It was predicted that such an approach could be used to drive an AFM to FM transition concurrent with a topological phase transition, using experimentally accessible ultrafast modalities. The magnetophononic wavemixing in the present work provides an experimental foundation for such approaches and a path toward achieving ultrafast lightinduced topological phase transitions.
Methods
Crystal growth and characterization
Single crystals of MnBi_{2}Te_{4} were grown using a selfflux method^{11}. Mixtures of 99.95% purity manganese powder, 99.999% bismuth shot, and 99.9999+% tellurium ingot with a molar ratio Mn:Bi:Te = 1:10:16 were loaded into an aluminum crucible and sealed in evacuated quartz tubes. The mixture is heated upto 1173 K for 12 h and slowly cooled down at the rate of 1.5 K/h to 863 K. This is followed by centrifugation to remove excess flux. The phase and crystallinity of the single crystals were checked by Xray diffraction. The antiferromagnetic order with the Néel temperature of 24 K was confirmed using SQUID magnetometry.
Raman spectroscopy measurements
Temperaturedependent Raman spectra were collected using a Horiba LabRam HR Evolution with a freespace Olympus BX51 confocal microscope. A 632.8 nm linearly polarized HeNe laser beam was focused at normal incidence using a LWD 50× objective with a numerical aperture of 0.5, with the confocal hole set to 100 μm. A Si backilluminated deep depleted array detector and an ultralowfrequency volume Bragg filter were used to collect the spectra, dispersed by a grating (1800 gr/mm) with an 800 mm focal length spectrometer. The system was interfaced with an Oxford continuousflow cryostat for lowtemperature measurements, using liquid helium as the cryogen.
Fielddependent magnetoRaman spectra were collected using a homebuilt Raman spectrometer. A 632.8 nm linearly polarized HeNe laser beam was focused at normal incidence using a LWD 50x objective with a numerical aperture of 0.82. A Si backilluminated deep depleted array detector and a set of ultralowfrequency volume Bragg filters were used to collect the spectra, dispersed by a grating (1800 gr/mm) with a 300 mm focal length spectrometer. The system was interfaced with an Attocube AttoDRY 2100 closedcycle cryostat for lowtemperature, high magneticfield measurements, using liquid Helium as the cryogen. The fieldinduced Faraday rotation in the objective was calibrated and corrected using a halfwaveplate.
The laser power was maintained below 50 μW in all measurements, to minimize laser heating and maintain the power well below the damage threshold. Laser heating was calibrated by measuring Raman phonon peak shifts as a function of and using thermal conductivity values from reference^{14}. Polarized spectra were obtained using a halfwaveplate to rotate the polarization of the incident beam, with a fixed analyzer.
After peak assignment using polarization analysis, temperature and fielddependent spectra were collected without a polarizer, to maximize signal throughput. Spectra were averaged over 60 and 120 min in the case of temperaturedependent and fielddependent measurements respectively, with a temperature stability of ±0.1 K. Any subtle drift in the spectrometer (<0.15 cm^{−1}) over the temperaturedependent studies was corrected using the HeNe line at 632.8 nm.
The A_{1g}^{(1)} peak was fit using an inverse Fano lineshape in combination with a linear background. Its lineshape is given by the expression \(\left(\omega \right)\,=\,\frac{{\left(q\Gamma \,\,\left(\omega \,\,{\omega }_{0}\right)\right)}^{2}}{{\Gamma }^{2}\,+\,{\left(\omega \,\,{\omega }_{0}\right)}^{2}}\), where I is the scattering intensity, ω is the energy, \({\omega }_{0}\) and Γ are the resonant energy and linewidth of the excitation respectively, and \(1/q\) is a measure of the peak asymmetry. The E_{g}^{(2)}, and A_{1g}^{(3)} peaks were fit with a standard Gaussian lineshape, and the E_{1g}^{(3)} and A_{1g}^{(2)} peaks were fit with a standard Lorentzian lineshape.
A nonlinear leastsquares fitting procedure was used. To ensure robustness of the temperaturedependent fits, the same initial fit values and constraints were used for each set of temperaturedependent and fielddependent spectra.
Magnetic fielddependent ultrafast optical spectroscopy
Ultrafast optical pumpprobe measurements were carried out using a 1040 nm 200 kHz SpectraPhysics Spirit Ybbased hybridfiber laser coupled to a noncollinear optical parametric amplifier. The amplifier produces <50 fs pulses centered at 800 nm (1.55 eV), which is used as the pump beam. The 1040 nm (1.2 eV) output is converted to white light, centered at 1025 nm with a FWHM of 20 nm, by focusing it inside a YAG (Yttrium Aluminum Garnet) crystal. The white light is subsequently compressed to ~50 fs pulses using a prism compressor pair and is used as the probe beam. The pump and the probe beams are aligned to propagate along the [001] axis of the crystal, at near normal incidence.
The samples were placed in a magnetooptical closedcycle cryostat (Quantum Design OptiCool). Pumpprobe measurements were carried out as a function of magnetic field applied normal to the sample surface (along the [001] direction). The sample temperature was fixed at 2 K. A pump fluence of ~100 µJ/cm^{2} was used in order to generate sufficiently large coherent phonon oscillations, while keeping the transient heating to a minimal amount, to ensure we avoid melting of the magnetic order.
Ultrafast electron diffraction measurements
Ultrafast electron diffraction measurements were carried out at the MeVUED beamline at the SLAC National Accelerator Laboratory. The principle and other technical details of the experimental setup are outlined elsewhere^{39}. A 60fs laser pulse with a photon energy of 1.55 eV and fluence of 7 mJ/cm^{2} were used to excite the sample. A higher pump fluence was required than in the optical pumpprobe measurements, in order to produce a sufficiently large pumpinduced change in diffraction intensities. Fluencedependent damage studies revealed no signs of laserinduced damage, and the measurements were repeatable over thousands of cycles. Femtosecond electron bunches of ~100 fs pulsewidth and 3.7 MeV kinetic energy were used to measure pumpinduced changes in electron diffraction intensities.
Measurements were carried out on flakes with an average thickness of around 100 nm, exfoliated from a single crystal of MnBi_{2}Te_{4} and transferred onto an amorphous Si_{3}N_{4} membrane using an exsitu transfer stage. The flakes were protected with an additional layer of amorphous Si_{3}N_{4} to prevent degradation. The spot sizes of the pump and probe beams were 464 × 694 µm and ~70 µm, respectively, and the measurements were carried out at 30 K.
The ultrafast electron diffraction intensities were obtained by averaging over several scans, normalizing individual diffraction images to account for electron beam intensity fluctuation. Individual diffraction peaks were fit to a twodimensional Gaussian function, and then averaged over symmetryrelated peaks based on the R3m space group of MnBi_{2}Te_{4}.
Pumpprobe data analysis
The timeresolved reflectivity traces were first fitted to a product of an error function and a biexponential decay function. The error function models the excitation of photocarriers and instrumental temporal resolution, and the exponential decay is an approximation for the sum of various unknown processes occurring over the measured time delay, including electronelectron and electronphonon thermalization. The functional form is:
where \(t\) is the time delay, \({\tau }_{{el}}\) is the rise time for the excitation of photocarriers, \({\tau }_{1}\) and \({\tau }_{2}\) are the time constants of exponential decay, and \({A}_{1}\), \({A}_{2}\), and C are constants.
Upon subtracting the biexponential decay, the residual traces were fit to the sum of two decaying sinusoidal functions. The functional form is:
where t is the time delay, \({f}_{1}\) and \({f}_{2}\) are the frequencies of the sinusoidal functionals, corresponding to the A_{1g}^{(1)} and A_{!g}^{(2)} phonons, \({\phi }_{1}\) and \({\phi }_{2}\) are the phases, and \({\tau }_{d1}\) and \({\tau }_{d2}\) are the time constants of exponential decay of the oscillations. The initial amplitudes \({A}_{1}\) and \({A}_{2}\) are plotted in Fig. 4d.
The ultrafast electron diffraction intensities were fit to an exponential decay function of the form:
where t is the time delay, \({\tau }_{l}\) is the time constant, and \({A}_{1}\) and C are constants.
Electronic structure and phonon calculations
DFT calculations were carried out using the Vienna Ab Initio Simulation Package (VASP)^{40,41,42,43,44} with the PBE exchange correlation functional^{45} and van der Waals correction via the DFTD3^{46,47} method with BeckeJonson damping. A Hubbard U was also added to the Mn (4 eV) using Dudarev’s^{48} approach. A nonprimitive cell containing two Mn atoms was used to obtain the equilibrium geometry of the system with AFMA magnetic structure. Γpoint phonons were obtained with the finite displacement method on a 1 × 1 × 1 “supercell” using the PHONOPY software package^{49} and VASP. An energy cutoff of 300 eV was used for all calculations. A 4 × 4 × 4 Γcentered kpoint mesh was used for equilibrium relaxations and phonon calculations. The general energy convergence threshold was 1 × 10^{−8} eV and the force convergence threshold for relaxation was 1 × 10^{−5} eV/Å. When including SOC in the magnetic parameter calculations, however, the energy convergence threshold was 1 × 10^{−6} eV. Gaussian smearing with a 0.02 eV width was also used in all relaxation and singlepoint energy calculations. Density of states calculations employed the tetrahedron method. The metallic state was modeled by electron doping the unit cells with 0.1 electron/Mn atom. Supercells for magnetic exchange calculations were generated using VESTA^{50}.
Exchange coupling constants calculations
Magnetic exchange parameters were obtained by considering a model spin Hamiltonian of the form \(H\,=\,\mathop{\sum}\limits_{\left\langle {ij}\right\rangle }{J}_{{ij}}{S}_{i}\,\cdot\ {S}_{j}\), where \({J}_{{ij}}\) includes intralayer exchange parameters J_{1} and J_{2} and interlayer exchange parameter \({J}^{\perp }\). A \(\sqrt{2}\) × \(\sqrt{2}\) × 1 supercell of the conventional cell was used get the intralayer exchange parameters, while a 1 × 1 × 2 supercell of the primitive cell was used to get the interlayer exchange parameter. Γcentered kpoint meshes of 4 × 4 × 1 and 4 × 4 × 4 were used in the respective calculations.
For the intralayer exchange parameters, one FM and two AFM configurations (stripe and upupdowndown) were used. The spin exchange energy equations in terms of magnetic exchange parameters for structures of \(R\bar{3}m\) symmetry are as follows:
For the interlayer exchange parameter, one FM and one AFM configuration were used.
The calculated values were multiplied by S^{2} to obtain the exchange coupling in meV, assuming the spin of the local moment is S = 5/2.
Data availability
Supplementary Information is available for this paper. All the data generated in this study have been deposited in the Figshare database at https://doi.org/10.6084/m9.figshare.19102934.v1.
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Acknowledgements
H.P., V.A.S., H.W., P.K., M.P., N.Z.K., A.M.L., R.A., J.M.R., and V.G. acknowledge support from the DOEBES grant DESC0012375. H.P. acknowledges partial support from the DOE Computational Materials program, DESC0020145. Support for crystal growth and characterization was provided by the National Science Foundation through the Penn State 2D Crystal ConsortiumMaterials Innovation Platform (2DCCMIP) under NSF cooperative agreement DMR1539916 and DMR2039351. D.P. was supported by the Army Research Office (ARO) under grant no. W911NF1510017. SLAC MeVUED is supported in part by the DOEBES SUF Division Accelerator & Detector R&D program, the LCLS Facility, and SLAC under Contract Nos. DEAC02–05CH11231 and DEAC02–76SF00515. Use of the Center for Nanoscale Materials, a DOE Office of Science User Facility, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DEAC0206CH11357. Zerofield Raman measurements were performed in the Materials Characterization Laboratory within the Materials Research Institute at Penn State.
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H.P. and V.G. conceived the project. Raman spectroscopy measurements and analysis were carried out by H.P., H.W., M.W., and V.G. Pumpprobe reflectivity was carried out by P.K., M.P., H.P., R.S., and R.A., and the results analyzed by H.P., P.K., M.P., and V.A.S. DFT calculations were done by N.Z.K., D.P., M.G., and J.M.R. Ultrafast electron diffraction was carried out by H.P., V.A.S., H.W., X.S., A.H.R., A.M.L., and X.W., and the results were analyzed by H.P. Crystal growth and characterization were done by S.H.L. and Z.Q.M. The paper was written by H.P. with inputs from all authors.
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Padmanabhan, H., Poore, M., Kim, P.K. et al. Interlayer magnetophononic coupling in MnBi_{2}Te_{4}. Nat Commun 13, 1929 (2022). https://doi.org/10.1038/s41467022295455
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DOI: https://doi.org/10.1038/s41467022295455
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