Distinct magneto-Raman signatures of spin-flip phase transitions in CrI3

The discovery of 2-dimensional (2D) materials, such as CrI3, that retain magnetic ordering at monolayer thickness has resulted in a surge of both pure and applied research in 2D magnetism. Here, we report a magneto-Raman spectroscopy study on multilayered CrI3, focusing on two additional features in the spectra that appear below the magnetic ordering temperature and were previously assigned to high frequency magnons. Instead, we conclude these modes are actually zone-folded phonons. We observe a striking evolution of the Raman spectra with increasing magnetic field applied perpendicular to the atomic layers in which clear, sudden changes in intensities of the modes are attributed to the interlayer ordering changing from antiferromagnetic to ferromagnetic at a critical magnetic field. Our work highlights the sensitivity of the Raman modes to weak interlayer spin ordering in CrI3.

induced phase transition when the interlayer spin arrangement changes from AFM to FM. Based on our results, we theoretically examine three origins of the new modes, including zone-folded phonons, bound two-magnons, or a bound state of a phonon and a soft magnon. This work validates magneto-Raman spectroscopy as a sensitive technique to probe interlayer magnetic ordering in quantum materials.
A thin piece of CrI3 (≈7 nm from atomic force microscopy, or ≈10 layers) was encapsulated between two 20 nm to 30 nm flakes of hBN using the dry transfer technique. 21,22 In the ab plane, the Cr 3+ atoms are arranged in a honeycomb lattice, where each Cr atom is bonded with six I atoms to form a distorted octahedron (see Figure 1a). The bulk Tc is around 61 K, 23 with the spins aligned perpendicular to the ab plane and the interlayer magnetic stacking is AFM, as demonstrated through a variety of experimental techniques. 2,8,9,[12][13][14][15]24,25 Raman spectra were collected with a triple grating spectrometer using a laser wavelength of 632.8 nm and keeping the power below 150 μW at the sample to avoid heating. The laser polarization of the incoming (εi) light (λ=632.8 nm) makes an angle φ with respect to the b axis ( Figure 1a), and the scattered (εs) light angle θ is changed from θ = 0° (parallel, xx) to θ = 90° (perpendicular, xy). Figure 1b shows the Raman spectra at T = 5 K, in both xx and xy configurations. It has been theoretically [26][27][28][29] and experimentally 25,30 suggested that atomically thin CrI3 does not go through the crystallographic phase transition, but instead remains in the monoclinic structure at low temperatures, resulting in AFM interlayer stacking. We confirm the monoclinic symmetry in our ≈10 layer flake by resolving two peaks at 108 cm -1 and 109 cm -1 between xx and xy, unlike the doubly-degenerate peak seen in the rhombohedral structure. 30 Thus, we label the phonons using the irreducible representations of the 2/m point group, where only the and modes are Raman active. 31,32 Two new modes appear below Tc in the xy at 77 cm -1 (9.5 meV) and 126 cm -1 (15.6 meV), labeled 1 and 2 ( 1,2 ) in Figure 1b, respectively. These modes were previously attributed to magnon excitations since they appear in the magnetically ordered state and have their largest intensity in xy (inset of Figure 1b), indicating symmetry. 33 Instead, the magnon dispersion at 5 K 34 shows a low-energy magnon at Γ below 1 meV (8 cm -1 ), similar to what was measured by recent FM resonance (FMR) experiments, 35 and magnons at the M point of the Brillouin zone at ~9 meV and 15 meV. Furthermore, a recent Raman study of magnon excitations in FePS3 showed it is possible for magnons to be present in both xx and xy. 20 We studied the effects of an applied magnetic field on 1,2 , as detailed in Figure 2. Two spectral ranges from 65 cm -1 to 90 cm -1 ( 1 , 1 ) and 120 cm -1 to 136 cm -1 ( 2 , 6 ) are shown on different intensity scales for clarity. At B = 0 T and in xy, 1 and 2 have strong intensities, whereas the two modes at slightly higher frequency have minimum intensities. Increasing the magnetic field results in drastic changes in the Raman spectra, where 1,2 behave in the same fashion. Above ≈1.6 T, the intensities of 1,2 abruptly start to vanish and 1 and 6 begin appearing in xy. By B = 2 T, 1,2 are absent in all polarization configurations, while 1 and 6 are no longer forbidden in xy. No further changes occur above 2 T, and no hysteresis was observed when the field was lowered back to 0 T. It should be noted that 1,2 do not show frequency shifting with magnetic field, suggesting they are not single magnons with spins perpendicular to ab.
Raman spectra were collected as the magnetic field was increased in finer steps below 2 T. This is shown as a false-color map ( Figure 3a) and spectra (Figure 3b) for the frequencies near 2 and 6 , where five distinct magnetic field ranges are revealed. Upon close inspection, there is an additional peak, which we label 1 , between 2 and 6 at approximately 128 cm -1 . No changes to 2 or 6 were observed in Range 1 from 0 T and 0.6 T. In Range 2 between 0.7 T and 0.8 T, 1 appears to increase in frequency. The spectra are stable through Region 3 from 0.8 T to 1.4 T, after which striking changes are seen in Regions 4 and 5. In Region 4, 2 starts to decrease in intensity and 1 decreases in frequency, but the intensity of 6 stays relatively constant.
Finally, in Region 5, 2 and 1 both decrease in intensity until they disappear, while 6 grows in intensity until ≈ 2 T, when the phase transition is complete.
Recent magneto-tunneling measurements of CrI3 also observed large changes in the tunneling current at nearly the same magnetic field values where we observe dramatic changes in the Raman spectra, such as at 0.8 T and 2 T. [12][13][14]24 These changes were attributed to the spinfiltering effect when the magnetic field is strong enough to change the interlayer spin arrangement from AFM to FM. The striking resemblance in jumps/plateaus at the same magnetic field values observed with magneto-tunneling to those herein with Raman spectroscopy implies Raman spectroscopy is detecting the phase transition caused by layers flipping spins from AFM to FM. The first jump at 0.8 T is most likely due to the surface layers (adjacent to the hBN) flipping while the second, final jump is the flipping of the internal layers at ≈2 T. In particular, mode 1 is extremely sensitive to the spin-flipping, displaying strong frequency shifts and intensity variations where the spin flips occur.
The polar plots for 6 at various magnetic fields in Figure 4a further verify the magnetic phase transition from AFM to FM interlayer stacking. While one part of the phase transition seems to occur between 0.7 T and 0.8 T, with likely the surface layers flipping spins, the polar intensity plot of 6 remains unchanged as not enough FM is introduced. For 2 T and 9 T, the polar plot of 6 is rotated by 30°. We know our sample is FM at high magnetic fields (B > 7 T) as we observe the low energy FM magnon 34,35 ( Figure S2). Thus, the similarity of the polar plots at 2 T and 9 T implies the phase transition at 2 T spin-polarizes the CrI3. Although no structural change took place (see Figure S3), the magnetic point group changes, resulting in new Raman tensor elements and thus new polar plots (see Supplementary Information).
The temperature dependence of the spin-flips was investigated by tracking the intensity of 2 as a function of magnetic field for different temperatures between T = 9 K and 26 K, as detailed in Figure 4b. The intensity of 2 is shown relative to the intensity of the combination peak 5 3 ⁄ at ≈ 115 cm -1 at B = 0 T for each temperature. The spin-flip transition field, or the amount of magnetic field necessary to cause 2 to disappear, decreased as temperature is increased. In addition, the distinct jumps from spin flips and flat plateaus observed in the intensity of 2 are smoothed out for higher T. Further temperature dependence is analyzed in Figure S4.
We consider three theoretical models for origins for 1,2 , which appear in the AFM state and then disappear in the FM state. Our data strongly suggests they are not pure one-magnon excitations with the spins normal to the ab plane, as 1,2 do not shift with applied magnetic field.
One possible explanation is that they are zone-folded phonons due to a doubling of the AFM unit  Table S1). Two pairs of modes with similar frequencies to 1 / 1 and 2 / 6 have an easily resolvable frequency splitting, with the zone-folded phonon ( 1 or 2 ) between 2 cm -1 and 4 cm -1 lower in frequency than the original phonon ( 1 or 6 ). These zone-folded phonons would not shift in magnetic field and would be Raman-active (forbidden) in the AFM (FM) state, aligning with the behaviors of 1,2 . While this theory is consistent with the symmetry of the modes for an even number of layers, where inversion symmetry is broken with AFM stacking, it cannot account for the presence of 1,2 in an odd number of layers, as is suggested by the data in Jin et al. 33 We also considered that the modes could be two-magnon bound states with a total spin of zero. Recent theoretical 36 and experimental 35 works suggest that there is a strong Kitaev interaction in CrI3. If this is true, it is predicted 37 that it costs less energy to create a bound state of two-magnons than to create a single spin-flip magnon, and thus two-magnon bound states would be favored. From the Hamiltonian of a CrI3 monolayer, we find that the two-magnon density of states (DOS) and the Raman spectrum shift with magnetic field (see Supplementary   Information), and thus intralayer two-magnon bound states cannot explain 1,2 .
The final model considered is that 1,2 are attributed to the bound state of a phonon (i.e. 1 for 1 , and 6 for 2 ) and a soft (low energy) magnon of the AFM structure, as detailed in the Supplementary Information. Here, the inelastic Raman scattering involves the excitement of a phonon and annihilation of a magnon, in which the bound state frequency is: where is the binding energy. The difference in 1 and 2 would result in a different splitting of 1 and 1 vs. 6 and 2 , as is observed in our results. In addition, this theory also explains the identical behavior of 1,2 with field, as both states involve the same AFM magnon that would be suppressed above 2 T in the FM state. The soft magnon (and hence the phononmagnon bound state) carries representation, agreeing with 1,2 exhibiting the highest intensity in xy. For these modes to not shift in a magnetic field, however, it is necessary for the layered AFM to have the spins point in the ab-plane, which we refer to as , as opposed to the claims of other reports. 2,8,[12][13][14]24 We note that the associated magnetic group 2´/m for , generated by magnetic rotation 2 ′ and mirror , is consistent with the angle-dependence of SHG signals observed in bilayer CrI3, 25 since the mirror and 2-fold rotation enforces similar constraints on the non-linear susceptibility with respect to in-plane polarization. It is also predicted tensile strain, such as substrate-induced strain, can flip the spin orientation to inplane. 38 Of note, our proposed models for the origins of 1,2 predict they would disappear for a monolayer sample. Although W. Jin et al 18 stated they observed 1,2 in a monolayer, their data shows no difference in frequency of the features in xx and xy polarizations, which strongly suggests that, for a monolayer, they only observed the 1 and 6 phonons.
In conclusion, we utilized magneto-Raman spectroscopy to elucidate a magnetic phase transition in CrI3 where the interlayer stacking changes from AFM to FM. Substantial changes in the Raman spectra are detected at specific magnetic field values due to spin-flips of layers to a FM state, indicating that Raman modes are extremely sensitive to this phase transition.
Moreover, Raman scattering is shown to be key to understanding the symmetry and frequency shifts of the modes. In terms of the new modes 1 and 2 , we conclude it is unlikely they are high frequency magnons as previously assigned, and examine three other possible explanations for their origins, including zone-folded phonons, bound two-magnons, and bound magnonphonon states. While these theories capture much of the behavior observed herein, they also contradict certain aspects of previously published work. Thus, this study paves the way for further theoretical and experimental studies of CrI3 to determine the details of the magnetic structure and the effects of encapsulation.

DFT Phonon Calculations:
We perform density functional theory (DFT) calculations 5,6 with the Quantum Espresso 7 code, using the GBRV ultrasoft pseudopotential set. 8,9 We use the vdw-df-ob86 10,11 exchange correlation functional, which includes longe range van der Waals interactions, for our main results. We also tested the PBEsol 12 functional, finding similar results. Phonon calculations were performed using a finite differences (frozen phonon) approach, using PHONONPY 13 to perform symmetry analysis and the cluster_spring 14 code to calculate phonon dispersions. A 6 x 6 x 4 k-point sampling was used for the ferromagnetic unit cell. † Certain commercial equipment, instruments, or materials are identified in this manuscript in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment are necessarily the best available for the purpose.  CrI3 that blueshifts with increasing magnetic field. We extract g ≈ 2.08 ± 0.05 and =0 = 2.21 ± 0.3 cm -1 (0.27 ± 0.04 meV).

Figure S3:
Comparing Raman spectra at B=9 T and T=9 K in the parallel (xx, black) and cross (xy, red) polarization configurations. Since we still observe a splitting in the mode at ≈108 cm -1 , as opposed to one degenerate mode, we deduce the thin CrI3 is still in the monoclinic phase after the magnetic phase transition from antiferromagnetic to ferromagnetic interlayer coupling. Figure S4: (b) Temperature dependence of the Raman spectra at B=0 T in xy polarization configuration, where P2 disappears above 32 K and the 6 mode remains small. (c) Temperature dependence at B=1.4 T, where increasing the temperature results in a similar evolution of the Raman spectra as further application of magnetic field, including the appearance/increased intensity of 6 , until it disappears again as the sample is warmed. Figures 4b and 4c demonstrate that temperature can also replace an increasing magnetic field to complete the phase transition in thin CrI3. Figure 4b shows the evolution of the Raman spectra at B = 0 T, in xy configuration, as a function of temperature. As the temperature of the CrI3 is increased, the intensity of 2 decreases until it is unobservable above T = 35 K. In this case, with no magnetic field applied, the intensity of 6 remains small, as expected. On the contrary, when a magnetic field slightly below is applied, such as B = 1.4 T in Figure 4c, increasing the temperature can be used in the same fashion as increasing the magnetic field to push CrI3 through the phase transition, as was shown in magneto-resistance as well. 15 In this case, a ≈16 K increase in temperature is enough to drive the phase transition. With the magnetic field applied, the intensity of 6 increases with increasing temperature, but as the temperature increases further towards the Curie temperature, 6 starts to disappear again, confirming that the phase transition we are observing is magnetic in nature. Figure S5: Raman spectra for bulk (≈mm thickness, not encapsulated in hBN) flake of CrI3 at T = 5 K. The phonons are labeled using the rhombohedral point group 3 ̅ , since bulk CrI3 does undergo a phase transition from monoclinic to rhombohedral at ≈200 K. Modes 1 and 2 were not observed, and in agreement with the rhombohedral structure, the mode at ≈107 cm -1 is degenerate in xx and xy polarization configurations.

Raman Tensors of CrI3
If the spins are aligned perpendicular to the ab plane, due to the presence of both the magnetoelectric effect 16,17 and second-harmonic generation 18 in monoclinic, bilayer CrI3, we deduce the magnetic point group in this AFM state (or at B < 2 T) to be 2 ′ ⁄ . The corresponding Raman tensors when the laser polarizations are in the ab plane are: 19 For FM interlayer exchange (magnetic field > 2 T), where the magnetic point group is now 2 ′ ′ ⁄ , the Raman tensors are:

= ( )
If we also consider that the tensor elements can be complex, as is the general case for absorbing materials, the relevant Raman tensors for AFM and FM interlayer exchange interactions are: and as Zone-Folded Phonons: DFT Calculated Phonon Frequencies Figure S6: Calculated phonon dispersion of (a) ferromagnetic and (b) antiferromagnetic stacking in bulk CrI3 in the monoclinic crystal structure. The relevant Brillouin zone is shown on the right from Bilbao crystallographic server. [20][21][22] In the AFM case, the unit cell doubles along the c-direction in real space, such that the A-point folds into Γ and the number of modes is doubled compared to the FM case. The symmetry of the Raman-active modes is marked as or at the Γ point.  Table S1: DFT-calculated phonon frequency and Q-points for the Raman-active phonons in monoclinically-stacked, bulk CrI3 for ferromagnetic (FM) and antiferromagnetic (AFM) interlayer exchange coupling. For FM exchange, we list the phonon frequencies and symmetries at the A-point in the Brillouin zone as well, which is along the c-direction in real space. In the AFM state, the unit cell doubles and the A-point is folded into Γ and thus can become Raman-active. In general, the splitting between the mode originally at Γ and the mode zone-folded from the A-point are very small and most likely not resolvable. However, for the pairs of modes highlighted in red, the splitting is significant, with the mode from A at a lower frequency than the one from Γ. These lower frequency, zone-folded modes could explain the presence of 1 and 2 in the AFM state, and why they disappear when the material has FM stacking. Figure S7: DFT-calculated phonon vibrations for 1 and 6 (phonons #13 and #33, respectively) as well as the zone-folded phonons at lower frequency. Eigenvectors are calculated for a bulk system, but are pictured as a bilayer system. Looking at the vibrations for a bilayer system, the two zone-folded vibrations would have symmetry with the inversion center in-between the two layers. When magnetic ordering is considered, however, where the two layers have AFM stacking, there is no longer inversion symmetry and the modes will be of symmetry, and thus can be Raman active. This would not, however, explain why 1 and 2 modes are seen in odd number of layer flakes, as suggested by Jin et al., 23 since inversion symmetry is preserved and thus these modes should remain Raman in-active.

and as Bound Two-Magnon Excitations:
We considered the possibility that 1 and 2 could be intralayer bound two-magnon excitations with a total spin of zero given that they do not shift with B field. In ferromagnetic systems, two-magnon excitations are expected to naturally arise from bond-dependent interactions like the Kitaev interaction, 24 which a recent experiment 25  We performed an exact diagonalization calculation of the two-magnon density of states (DOS) and Raman intensity at zero temperature for a spin-3/2 system of 6-sites (a single honeycomb plaquette) described by the JKΓ Hamiltonian that the entire two-magnon DOS and Raman spectrum (see Figure S8) shift with applied field, thereby ruling out this potential mechanism for 1 and 2 . Figure S8: Exact diagonalization calculation of the zero-temperature Raman spectra in the cross (xy) polarization configuration for a spin-3/2 system of 6-sites (a single honeycomb plaquette) described by the JK Hamiltonian and using the values of the coupling constants obtained by Lee et al. 25 This calculation serves as an estimate of the Raman behavior of monolayer CrI3. When a field is applied out of the plane, the Raman spectrum shifts.

and as Bound Magnon-Phonon Pairs:
We propose that, if there was in-plane antiferromagnetic order as reported in CrCl3, 1 and 2 could correspond to the bound state of a phonon at a slightly higher frequency ( 1 for 1 , and 6 for 2 ) and a soft magnon (0,0, ),+ . This magnon (and hence the phonon-magnon bound state) carries representation of the 2 ′ / magnetic group in our 10-layer sample and exhibits the highest intensity in cross polarization. The associated magnetic group 2 ′ / generated by magnetic rotation 2 ′ and mirror is also consistent with the angle-dependence of second harmonic generation (SHG) signals observed by Z. Sun et al. 18 , since mirror and 2-fold rotation 2 enforces similar constrains to nonlinear susceptibility with respect to in-plane polarization. When the perpendicular magnetic field exceeds the critical value of 2 T, the in-plane order is destabilized and transitioned into out-of-plane ferromagnetism at larger fields.
We use the isotropic Heisenberg interaction between nearest neighbors as a minimal model in order to extract the symmetry properties of the magnon modes. The acquired representation of magnon modes and selection rules from Raman scattering are expected to be universal, insensitive to the exact form of the Hamiltonian. However, the frequency of the magnon modes can be an artifact of such toy models and should not be compared directly with experiments.
The monoclinic space group 2/ is generated by the following point group symmetries: and their combination is the inversion symmetry.
The following minimal model of spin-3/2's is considered: where ≫ , > 0 are all positive parameters.

I. LARGE FIELD CASE: OUT-OF-PLANE FERROMAGNETISM
For ferromagnetism (FM) along -axis (coined ), we use the following Holstein-Primakoff representation to derive the spin wave theory: In this phase, the unbroken magnetic symmetries (magnetic point group 2 ′ / ′ ) are where T is the time-reversal operator. The magnon operator transforms under them as

II. SMALL FIELD CASE: IN-PLANE ANTIFERROMAGNETISM
As suggested by the observation of SHG in bilayer CrI3, 18 multilayer CrI3 is likely to exhibit antiferromagnetism (AFM) between two neighboring layers. In the limit of a small magnetic field < 2 T, we consider an in-plane moment along the ̂-axis as reported in CrCl3. 26 The Holstein-Primakoff representation writes: For an even number of layers = 0 mod 2, the magnetic point group is 2 ′ / generated by 2 ′ and .
There are two soft magnon modes, (0,0,0),+ and (0,0, ),+ . Their symmetry characters are summarized in Table S2. They belong to and representations of group 2/ , and the mode The associated symmetry representations of magnons are summarized in Table S3. In this case, both soft magnons are Raman active and belong to representation.  Table S2. Symmetry characters (magnetic point group 2 ′ / ) of magnons in the phase, in a thin film of CrI3 with an even number of layers.

Modes
Irrep. Raman active?  Table S3. Symmetry characters (magnetic point group 2/ ) of magnons in the phase, in a thin film of CrI3 with an odd number of layers.
For comparison, we also list the representations of magnon modes at zone center Γ, for the case of even (Table S4) Table S4. Symmetry characters (magnetic point group 2/ ′ ) of magnons in the phase, in a thin film of CrI3 with an even number of layers.  Table S5. Symmetry characters (magnetic point group 2 ′ / ′ ) of magnons in the phase, in a thin film of CrI3 with an odd number of layers.