Introduction

Kagomé materials possess two-dimensional periodic networks of corner-sharing triangles and, as a result, exhibit a high degree of geometrical frustration. This structural frustration can create Dirac cones and flat bands in the electronic structure1,2,3,4,5 as well as exotic magnetic ground states6,7,8 accompanied by the anomalous Hall effect9,10,11,12, charge fractionalization13,14, chiral magnetism15,16, and strong electron correlations17. The discovery of charge density waves (CDWs) in superconducting AV3Sb5 (A = K, Rb, Cs) and magnetic FeGe demonstrates that CDWs can exist in both magnetic and nonmagnetic kagomé lattices across a range of electron correlations18,19,20. At the same time, rich phase diagrams can be obtained by tuning the frustration and electron filling in the kagomé lattice. Strong entanglements make this platform well-suited to revealing intertwined and competing states.

Recently, a family of bi-layer analogs with chemical formula RV6Sn6 (R = Sc, Y, Lu, Tb, Ho, Gd...) has attracted attention [Fig. 1a]21. These kagomé metals also host topological Dirac surface states, van Hove singularies, anisotropic magnetism, and other exciting properties22,23. Unlike all three AV3Sb5 compounds, superconductivity has not been reported in ScV6Sn6 under any temperature or pressure conditions investigated thus far24,25, possibly due to the lack of a Γ-centered Fermi pocket in this material26,27,28,29. Further, only ScV6Sn6 exhibits a three dimensional CDW, making it comparable - at least in some ways - to the AV3Sb5’s. Despite recent activity in this highly contemporary research area, there is much more to learn about the CDW in ScV6Sn6 and related materials. The combination of two kagomé layers per unit cell along with the lower symmetry CDW wavevector makes resolving structural details and the phase transition mechanism of ScV6Sn6 a particularly daunting task.

Fig. 1: Properties of hexagonal ScV6Sn6 at room temperature.
figure 1

a Crystal structure of ScV6Sn6 (space group P6/mmm, #191)25. This three dimensional material hosts two vanadium kagomé layers separated by ScSn2 and Sn2 layers along the out-of-plane direction. b Raman scattering spectrum of ScV6Sn6 compared with predictions from complementary lattice dynamics calculations. The spectrum is collected on the ab plane in the back-scattering geometry. The inset shows the displacement pattern of the A1g symmetry mode near 240 cm−1. c Close up view of the phonon near 240 cm−1 in the RV6Sn6 family materials (R = Sc, Y, Lu).

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ScV6Sn6 is a paramagnetic metal with a first-order CDW transition at 92 K25,30,31,32. The long-range ordered CDW involves primarily out-of-plane Sc and Sn displacements; thus far, the contribution of the V centers has been considered to be small25,33. First-principles calculations point to lattice instabilities (soft modes) in ScV6Sn633 - possibly due to the small size of the Sc3+ radius. Experimentally, short-range order has been observed in the high temperature phase with wavevector (1/3, 1/3, 1/2)31,32. Stable long-range CDW order with wavevector (1/3, 1/3, 1/3) sets in below TCDW = 92 K25,31,32. This type of lattice instability is not present in the Y and Lu analogs, and there are no CDWs in these materials33.

Experimental confirmation of these findings by other probes is highly desirable. Traditionally, x-ray and vibrational spectroscopies have been used to unravel CDW mechanisms. Infrared and Raman scattering spectroscopies in particular are well-suited for exploring the microscopic aspects of local lattice distortions as well as phase and amplitude modes of a CDW. Of course, the metallic character of ScV6Sn6 challenges this approach because the odd-symmetry infrared-active phonons are screened by the Drude contribution34. Raman scattering provides a way forward35, and it has been used to gain significant insight into CDWs in other kagomé systems36,37 despite the fact that it accesses only even-symmetry excitations at the zone center. Naturally, ScV6Sn6 has more Raman-active modes than members of the AV3Sb5 family due to the larger unit cell.

An open question about ScV6Sn6 is why it has a different CDW wavevector than its better studied AV3Sb5 counterparts. Even if the driving force of the CDW instability were the same in these two families of vanadate kagomés, variations can arise from the electron count as well as the overall number of atoms in the unit cell thus producing distinct electronic structure near the Fermi level38. Furthermore, the apical Sb ions are important in determining the relative stability of CDW phases in CsV3Sb539. This implies that the double-kagomé layer structure in ScV6Sn6 may impact the P vs. L point instabilities and, when they freeze in, their respective energy gains. Note that in this compound, instabilities both at the \(P(\frac{1}{3},\,\frac{1}{3},\,\frac{1}{3})\) and \(L(\frac{1}{2},\,0,\,\frac{1}{2})\) points are present. Both of these instabilities are comprised of mostly out-of-plane Sc displacements, and they lead to metastable phases with competing energies33. Hence, elucidating phonon behavior across the CDW transition is important to understanding CDW phase stability in this system.

In this work, we combine temperature- and pressure-dependent Raman scattering spectroscopy of ScV6Sn6 with complementary lattice dynamics calculations to reveal the properties of the charge density wave states in this model bi-layer kagomé metal. What distinguishes our work from prior efforts40 is the quality of our single crystals with different R site substitutions, the ability to employ both temperature and pressure as tuning parameters, and our symmetry-guided strategy of projecting the high temperature phase P6/mmm phonons onto those in the low temperature CDW state to uncover their origins. We find that the A1g symmetry mode near 240 cm−1 - which involves out-of-plane V center motion - is very sensitive to the development of the CDW. For instance, even though it corresponds to a singly-degenerate phonon mode, it seemingly displays five-fold splitting in the low temperature phase consistent with \(R\bar{3}m\) symmetry. We discuss this multiplet structure in terms of mixing of nearby symmetry-appropriate and zone-folded phonons, specifically those of A1g, Eg, and P symmetries. We also demonstrate that compression at room temperature may quench the recently reported short-range CDW in ScV6Sn6 but has no effect on the Lu analog. We therefore establish that density wave stability is impacted by both physical and chemical pressure. These findings revise our understanding of phonons and their mixing processes in ScV6Sn6 related materials.

Results and Discussion

Raman-active phonons of ScV6Sn6 at 300 K

Figure 1b summarizes the Raman scattering response of ScV6Sn6 at room temperature. We employ a symmetry analysis, complementary lattice dynamics calculations, polarizer/analyzer combinations, and chemical substitution on the R site to assign the excitations. Overall, the eight calculated phonon frequencies (blue spheres in Fig. 1b, see Supplementary Table 2) are in excellent agreement with the measured spectrum. Challenging assignments are near 240 cm−1 where there are two candidate modes and below 200 cm−1 where the Raman-active phonons are embedded in a series of extra peaks (Supplementary Fig. 1a). The latter turn out to be rotational modes of air in the optical path. They appear in this and other measurements of ScV6Sn640,41 due to the low brightness of the sample but can be omitted by purging the optical path and by using standard oscillator-fitting techniques as we did here. The peak near 240 cm−1 is important to our continuing discussion. According to our lattice dynamics calculations, there are two candidates for this structure: E1g and A1g symmetry modes. Surprisingly, the peak can be fit by a single oscillator suggesting that one of these constituents is present with extremely low intensity. Based upon extinction behavior under various polarizer/analyzer arrangements [Supplementary Fig. 1b], we assign the 240 cm−1 peak as primarily an A1g symmetry mode. The E1g mode is essentially undetectable due to its low intensity. The displacement pattern of the A1g symmetry mode, which involves out-of-plane V center motion, is shown in the inset of Fig. 1b. Substitution on the rare earth site is less useful for assignment purposes because the frequency vs. mass trend is not straightforward [Fig. 1c]. The atomic radius of the R center and the precise local environment appear to be more important. The complete set of mode displacement patterns in ScV6Sn6 is given in Supplementary Table 1.

Spectral changes across the CDW transition

Figure 2a summarizes the Raman scattering response of ScV6Sn6 as a function of temperature in the high frequency region. We focus on the behavior of the A1g symmetry mode near 240 cm−1. This structure hardens systematically with decreasing temperature and appears to split into a cluster of at least five closely-spaced peaks below 90 K. (The low brightness of this sample and resolution issues make it difficult to tell whether the small shoulders and tiny features on the baseline are real - sufficiently different from nearby peaks as well as the noise level.) This cluster is the most conspicuous signature of the CDW phase, although it obviously raises questions of exactly how and why a singly-degenerate vibrational mode might split in a low symmetry environment. We show the behavior of the E2g phonon near 300 cm−1 for comparison. This structure hardens anharmonically with decreasing temperature but does not split across TCDW - in line with the other Raman-active modes in ScV6Sn6 including those of A1g symmetry. Complementary measurements of LuV6Sn6 and YV6Sn6 reveal no low temperature fine structure near 240 cm−1 [Supplemental Fig. 2], consistent with transport results indicating the absence of CDW transitions in the R = Lu and Y analogs21,23. This demonstrates that the new cluster of peaks near the A1g symmetry mode is a signature of the unusual CDW state in ScV6Sn6.

Fig. 2: Symmetry breaking across the CDW transition in ScV6Sn6.
figure 2

a Summary of the Raman-active modes near 240 and 300 cm−1 as a function of temperature. All the spectra are collected on the ab plane in the back-scattering geometry. b Close-up view of the spectra at selected temperatures along with an oscillator fit. Several new features emerge in the CDW phase due to symmetry breaking. A minimum of five Voigt oscillators are needed to model the base temperature data. c, d Frequency vs. temperature and phonon lifetime vs. temperature trends for the features near 240 cm−1. These findings were extracted from the results in (b). In (c), error bars are on the order of the symbol size; a characteristic error bar is indicated in (d). e Close-up view of the E2g symmetry mode near 300 cm−1 as a function of temperature. This mode can be fit with a single Voigt oscillator over the full temperature range. e, f Frequency vs. temperature and phonon lifetime vs. temperature trends for the E2g symmetry mode near 300 cm−1. In (f) error bars are on the order of the symbol size; a characteristic error bar is shown in (g).

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We quantify these results by fitting the Raman scattering response of ScV6Sn6 with a series of Voigt oscillators and a linear baseline. Close-up views of the modes of interest along with their oscillator fits are shown in Fig. 2b, e. A single oscillator is required to fit the A1g symmetry mode in the normal state, whereas a minimum of five oscillators are needed to fit the spectra in the CDW state. By contrast, a single oscillator fits the E2g mode near 300 cm−1 over the full temperature range. The center frequency and full width at half maximum (FWHM) are extracted from these fits. The phonon lifetime τ is a useful quantity related to the Heisenberg uncertainty principle that can be calculated as τ = ℏ/FWHM42.

Figure 2c, d displays frequency and phonon lifetime trends as a function of temperature for the A1g symmetry phonon. The sharp peaks that develop immediately below TCDW are consistent with a first-order phase transition25,30,31,32, although as we shall see below, the multiplet structure is not due to symmetry breaking-induced splitting because a singly-degenerate vibrational mode cannot not split further as part of a group-subgroup transition. At the same time, the strong clustering seems to argue against the appearance of traditional zone-folded phonons unless the phonon bands are rather flat and fold into a similar frequency window37,43,44. Overall, the phonon lifetime rises from 1.25 ps at room temperature to between 2.2 and 4 ps in the low temperature phase depending on the branch.

Figure 2f, g displays frequency and phonon lifetime of the E2g phonon as a function of temperature. The mode hardening can be modeled by characteristic anharmonic effects45 where \(\omega (T)={\omega }_{0}+A(1+\frac{2}{{e}^{x}-1})+B(1+\frac{3}{{e}^{y}-1}+\frac{3}{{({e}^{y}-1)}^{2}})\) with x = ℏω0/2kBT, y = ℏω0/3kBT. Here, ω0 is the characteristic frequency at base temperature, A and B are constants. There are no anomalies in the frequency vs. temperature curve near the CDW transition. On the other hand, the phonon lifetime shows a pronounced kink at TCDW, rising sharply toward a limiting low temperature value of 2.5 ps. We carried out a similar analysis of the lower frequency phonons in ScV6Sn6 as well. No additional fine structure or unusual phonon softening was identified within our sensitivity, although we emphasize that ScV6Sn6 is a very low brightness sample. Thus, the new set of peaks near 240 cm−1 is the most reliable for unraveling phonon mixing in the CDW state.

Phonon mixing in the low temperature phase of ScV6Sn6

In order to better understand the multiplet structure of the 240 cm−1 phonon, we performed a symmetry analysis of the phonon modes46. In particular, we solved the subduction problem which relates the irreducible representations of the high-temperature P6/mmm structure and the low-temperature \(R\bar{3}m\) structure [Table 1]. This is a powerful technique that unravels a set of perturbed modes in terms of unperturbed vibrational modes47,48. As an aside, while we are aware that other studies have identified the CDW structure as belonging to the R32 space group25, our DFT results suggest \(R\bar{3}m\) is more likely (see Supplementary Information for additional discussion). As expected, A1g irreducible representations in P6/mmm, which exhibit the full crystal symmetry of the P6/mmm structure, also exhibit the full crystal symmetry of the subgroup \(R\bar{3}m\). They thus map to A1g in \(R\bar{3}m\) as well. This irreducible representation is Raman-active in both structures, but is only one-dimensional - a single A1g peak at 240 cm−1 in the high-temperature structure cannot, on its own, account for all five new modes in the \(R\bar{3}m\) structure. Where, then, do the new modes come from?

Table 1 Subduction relations between the high-temperature P6/mmm and low-temperature \(R\bar{3}m\) structures.
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The results in Table 1 reveal that new Raman active modes can appear in \(R\bar{3}m\) from the B1g modes of P6/mmm at the zone center, as well as P1 and P3 modes which fold in to Γ from the \(P\equiv (\frac{1}{3},\frac{1}{3},\frac{1}{3})\) point on the Brillouin zone boundary in reciprocal space (defined in Supplementary Table 3). In order to gain a quantitative understanding of the origin of each mode, we calculate the phonons in the \(R\bar{3}m\) structure (with frequencies given in Supplementary Table 5), then perform an analysis where the eigenvectors of the dynamical matrix associated with each phonon mode in the \(R\bar{3}m\) structure are projected onto the eigenvectors of the dynamical matrix in the P6/mmm structure as described in the Methods section. While all symmetry-allowed P6/mmm phonon modes are allowed to mix in the formation of the \(R\bar{3}m\) phonon modes (for example, a given \(R\bar{3}m\) A1g mode could exhibit significant contributions from multiple P6/mmm A1g, B1g, and P1 modes simultaneously), this technique allows us to understand the relative magnitudes of those contributions.

Figure 3 shows that eight Raman-active \(R\bar{3}m\) modes in the 228 to 241 cm−1 range exhibit significant overlap with A1g, E1g, P1, and P3 phonons in the 220 to 235 cm−1 range in the P6/mmm structure. Thus, the five clear peaks that abruptly appear alongside of the A1g peak at 240 cm−1 are likely a subset of these eight modes, and therefore originate not just from the A1g peak at 240 cm−1, but from E1g modes as well as modes at the P point in reciprocal space which fold into the Γ point and become Raman-active in the \(R\bar{3}m\) structure. In other words, we attribute the formation of multiplet structure in the CDW state to contributions from the A1g and E1g modes, which also undergo hybridization with P modes. It is noteworthy that the A1g mode near 240 cm−1 is unique as it is the only observable feature in the high temperature phase that also contributes to the formation of the five peaks in the CDW state. This indicates that the A1g phonon plays a peculiar and as-yet not fully-defined role in the process. One possibility is that the A1g mode couples strongly with the CDW modes, as it does in CsV3Sb539, and hence plays an important role in the stabilization of the P point CDW mode25.

Fig. 3: Overlap of eigenvectors of the dynamical matrix between phonons in the P6/mmm structure (vertical axis) and phonons in the \({{{\rm{R}}}}\bar{3}{{{\rm{m}}}}\) structure (horizontal axis).
figure 3

The color intensity of each box denotes the magnitude of the overlap, with a maximum possible amplitude of unity. The lefthand panel corresponds to projections of P6/mmm phonons transforming as A1g, B1g, and P1 onto \(R\bar{3}m\) phonons transforming as A1g, while the right-hand panel corresponds to projections of P6/mmm phonons transforming as E1g, E2g, and P1 onto \(R\bar{3}m\) phonons transforming as Eg. Taken together, these two panels include all Raman-active phonon modes in the \(R\bar{3}m\) structure, as well as all P6/mmm phonon modes related to them through subduction [Table 1]. Raman active modes in the P6/mmm structure are denoted with a “⋆” character. In both panels, the area of interest corresponding to phonons from 228 cm−1 to 241 cm−1 in the \(R\bar{3}m\) structure are boxed. Note that the data as presented need not be symmetric.

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Pressure destabilizes the CDW state

Figure 4 summarizes the Raman scattering response of ScV6Sn6 under pressure at room temperature. We focus on the behavior of the A1g symmetry phonon near 240 cm−1 because (i) it is most strongly connected to the CDW state and (ii) all other modes show only simple hardening under compression - at least within our sensitivity. As indicated in Fig. 4a, b, this peak can be modeled with a single oscillator below 2 GPa and also above approximately 6 GPa. At intermediate pressures, the spectra of ScV6Sn6 are best fit with two Voigt oscillators indicating the presence of a mixed phase. This type of mixed or two-phase regime is typical for a first-order pressure-driven transition49. In our case, the mixed phase consists of P6/mmm + a new high pressure phase. Based on this behavior, we define two critical pressures: PC,1 = 2 GPa and PC,2 = 6 GPa.

Fig. 4: Pressure-driven structural phase transition in ScV6Sn6 at room temperature.
figure 4

a, b Raman scattering response as a function of pressure. The dotted lines are experimental data, and the solid lines correspond to fitted curves. c, d Raman shift and phonon lifetime as a function of pressure for the 240 cm−1 phonon. Two critical pressures (PC,1 = 2 GPa and PC,2 = 6 GPa) separate the P6/mmm phase, the mixed phase, and the high pressure (HP) phase. The error bars in (c) are on the order of the symbol size; a characteristic error bar is indicated in (d).

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Figure 4c, d displays frequency vs. pressure and phonon lifetime trends for ScV6Sn6. We find that the concentration of P6/mmm (as measured by the relative strength of the low frequency oscillator) diminishes with increasing compression in the mixed phase region whereas that of the new high pressure phase (measured by the relative strength of the second peak) increases until it dominates above PC,2. The high pressure phase of ScV6Sn6 is very similar to but not identical to P6/mmm. The single A1g-like phonon mode emerging above PC,2 has a higher frequency than that in the P6/mmm ground state, pointing to a stronger force constant as well as a volume reduction in the high pressure phase which may have interesting connections to the inter-layer coupling in this class of kagomé metals. At the same time, the phonon lifetime drops from 1 ps at ambient pressure (lower than most traditional semiconductors and chalcogenides) to approximately 0.2 ps over this range. The decreased lifetime is due to significantly increased scattering events under compression42. We mention in passing that we could not stabilize the high pressure phase in our first principles simulations under pressure. This may be due to complications from the residual density wave state recently reported at room temperature31,32 or a shortcoming of density functional theory.

We postulate the following link between the pressure-driven transition in Fig. 4 and density wave stability in this material. One key fact is that recent transport work in the low temperature phase of ScV6Sn6 reveals complete suppression of the long-range CDW at approximately 2.4 GPa24. It is tempting to claim that this destabilization corresponds to the region between PC,1 and PC,2, but we must remember that our measurements are performed at room temperature - where the CDW is not fully developed. The recently reported short-range CDW builds upon this finding and provides a way forward. This “residual CDW” resides in the high temperature phase of ScV6Sn6 and hosts a (1/3, 1/3, 1/2) propagation vector31,32. As there are no high pressure structural phases stabilized in our first-principles simulations and no other orders present at room temperature in ScV6Sn6, we hypothesize that pressure quenches short-range CDW correlations in the high temperature phase of this system as well. It appears to do so in a first-order mixed phase manner49 between PC,1 and PC,2.

We tested this hypothesis using crystals with different rare earth centers. The crucial result is that similar high pressure Raman scattering measurements on the Lu analog reveal no evidence for the type of behavior observed in ScV6Sn6. Instead, the A1g symmetry mode near 240 cm−1 hardens systematically under compression [Supplementary Fig. 3] in line with the fact that LuV6Sn6 does not host a CDW50 and therefore does not host pressure-induced destabilization of a short range CDW at room temperature. These structure-property relations are nicely unified by R site size arguments [Inset, Supplementary Fig. 1c] which reveal that both chemical and physical pressure drive similar trends in CDW stability. In this scenario, the behavior of the A1g mode potentially signals the development of both long- and short-range CDW order.

There are few parallels between the pressure-induced transition in ScV6Sn6 and those observed in the single-layer vanadate kagomés like AV3Sb5. This is because the latter takes place at low temperature and between two different long-range-ordered CDW phases and the high symmetry phase, whereas in ScV6Sn6 this transition takes place at room temperature, well above the long-range CDW order39,51,52. An alternate possibility that accounts for the disappearance of the CDW at high pressures is that the system undergoes a volume collapse transition similar to those observed in the ThCr2Si2 family of compounds, which involves the formation of Sn–Sn covalent bonds53,54. Such transitions are also observed in various iron-pnictide superconductors, and couple strongly to superconductivity because of their effect on the Fermi surface55,56. While it may be too early for a full physical picture to emerge due to the limited number of measurements of ScV6Sn6 under compression, such a transition - if it exists - is likely to suppress the CDW as well. We note, however, that there is no evidence for Sn–Sn bond formation in our spectra, making this scenario less likely.

Unraveling CDW stability in the RV6Sn6 family of materials (R = Sc, Y, Lu)

In this family of materials, most of the focus so far has been on the electronic properties38,40,57,58,59. There has been significantly less effort to uncover vibrational contributions to the mechanism. This is partly because metallic character in both high and low temperature phases of ScV6Sn6 prevents us from revealing the behavior of the infrared-active phonons due to screening by the Drude peak34. The Raman scattering response is, however, still accessible35, and combined with different external stimuli and complementary lattice dynamics calculations, a remarkable picture of phonon mixing is beginning to emerge.

In this work, we traced how phonons mix in the CDW state of the double-layer vanadate ScV6Sn6, tested these ideas with symmetry arguments based upon our lattice dynamics calculations, and compared our findings to behavior in the Y and Lu analogs. Overall, we demonstrate that low temperature stabilizes the CDW whereas both physical and chemical pressures destroy it. This suggests that the electron-phonon interactions are on the “knife’s edge”, easily manipulated by any external stimuli and more sensitive than the similar CDW orders in the monolayer AV3Sb5 vanadates. Our analysis reveals that the multiplet character near 240 cm−1 in the CDW state of ScV6Sn6 is due to mixing of P6/mmm and \(R\bar{3}m\) phase phonons - a result that we quantify by projecting phonons of the high symmetry state onto those of the lower symmetry structure. We emphasize that these five peaks are not due to symmetry breaking of a single mode in the conventional sense. Instead, this multiplet is a combination of both Γ and P point modes that are folded to the zone center and mixed such that they become Raman active. This process indicates that the density wave state is influenced by a variety of phonon modes and highlights the intricate, phonon-assisted nature of the low temperature phase. Similar phonon hybridization has been observed in CsV3Sb537. Since the A1g symmetry phonon near 240 cm−1 may also serve to connect the long- and short-range CDWs, it provides a sensitive, microscopic indicator of CDW stability under different tuning parameters.

Methods

Crystal growth and diamond anvil cell loading

High quality RV6Sn6 (R = Sc, Lu, Y) crystals were grown from a Sn-rich melt method with a composition ratio of R:V:Sn = 1:6:60 as described in ref. 25. They are hexagonal blocks with clear ab surfaces. For the high pressure Raman scattering measurements, a small single crystal was loaded into a symmetric diamond anvil cell suitable for work in the 0–13 GPa range. The cell is equipped with low fluorescence diamonds with 400 μm culets. We also employed a stainless steel gasket with a 100 μm hole, KBr as the pressure medium, and an annealed ruby ball for pressure determination via fluorescence60.

Raman spectroscopy

Raman scattering measurements were performed in the back scattering geometry using a Horiba LabRAM HR Evolution spectrometer equipped with a 532 nm (green) laser, a 50 × microscope objective, 1800 line/mm gratings, and a liquid-nitrogen-cooled charge-coupled device detector. To minimize heating and maximize signal intensity for this low brightness sample, power was controlled below 7.7 mW, and the laser was slightly defocused. Each spectrum was integrated for 200 s and averaged four times. Variable temperature work was carried out with a low-profile open-flow cryostat, and high pressure measurements employed a diamond anvil cell as described above. Standard peak fitting techniques were employed as appropriate.

Lattice dynamics calculations

All density functional theory (DFT) calculations were performed with Projector Augmented Waves (PAW) as implemented in the Vienna Ab initio simulation package (VASP) version 5.4.461,62,63 using the PBEsol exchange-correlation functional for valence configurations of Sc, V, and Sn corresponding to 3s23p64s13d2, 3s23p64s13d4, and 5s24d105p2, respectively. Unless otherwise mentioned, experimentally determined lattice parameters of a = 5.475 Å and c = 9.177 Å for the P6/mmm structure and a = 9.456 Å and c = 27.412 Å for the conventional \(R\bar{3}m\) structure were used for all calculations Internal degrees of freedom were relaxed, with forces converged to within 0.001 eV/Å using a plane wave cutoff energy of 400 eV, combined with a Γ-centered Monkhorst-Pack k-point mesh of 20 × 20 × 10 in the P6/mmm structure, as well as a Gaussian smearing parameter of 10 meV. Calculations in the \(R\bar{3}m\) structure were carried out in a primitive, three formula-unit cell commensurate with the wavevector \({{{\bf{P}}}}=(\frac{1}{3},\frac{1}{3},-\frac{1}{3})\), which corresponds to a unit cell with basis vectors \((\frac{2}{3},\frac{1}{3},\frac{1}{3}),(-\frac{1}{3},\frac{1}{3},\frac{1}{3}),(-\frac{1}{3},-\frac{2}{3},\frac{1}{3})\), where the indices (a, b, c) correspond to the lattice vectors of the conventional \(R\bar{3}m\) nine formula unit cell. All computational parameters used for this structure were the same as for the P6/mmm structure, except with a Γ-centered Monkhorst-Pack k-point mesh of 10 × 10 × 10. To calculate phonon frequencies and their associated distortions, we constructed the dynamical matrix in a basis of symmetry adapted modes, which bring the dynamical matrix into block diagonal form, where each block corresponds to a single irreducible representation of the space group. These symmetry-adapted linear combinations of atomic displacements were found using the ISOTROPY software suite64.

Phonon overlap

After calculating the eigenvectors of the dynamical matrix associated with each phonon mode, we then computed the relationship between eigenvectors \({\hat{e}}_{i}^{I\alpha }\) in the P6/mmm structure (where I is an irrep, α the mode index within that irrep, and i the ionic degree of freedom in cartesian coordinates), and the eigenvectors \({\hat{g}}_{i}^{I\alpha }\) in the \(R\bar{3}m\) structure. First, the \({\hat{e}}_{i}^{I\alpha }\) were expressed in a three formula unit basis commensurate with the \(R\bar{3}m\) cell, then renormalized. The overlap for two phonons is then defined as the projection of \({\hat{g}}_{i}^{I\alpha }\) onto \({\hat{e}}_{i}^{I\alpha }\),

$$O(J,\beta ,I,\alpha )=\sqrt{{\left({\hat{g}}_{i}^{J\beta }{\hat{e}}_{i}^{I\alpha }\right)}^{2}}.$$
(1)

In the left hand panel of Fig. 3, J = A1g, and I ∈ A1g, B1g, P1, as determined by the solution to the subduction problem described in Table 1. In the righthand panel, J = Eg, and I ∈ E1g, E2g, P3. In this second case, the E1g and E2g phonons are doubly degenerate, and the P3 phonons quadruply degenerate. In order to compress the results of the table, instead of reporting a cell for each individual degenerate mode, our figure includes a single cell with amplitude defined as \(\tilde{O}\), where

$$\tilde{O}(J,\beta ,I,\alpha )\equiv \frac{1}{{M}_{\beta }}\mathop{\sum}\limits_{{\beta }_{l}}\sqrt{\mathop{\sum}\limits_{{\alpha }_{m}}O{(J,{\beta }_{l},I,{\alpha }_{m})}^{2}}.$$
(2)

Here, l and m run over the degenerate indices of β and α respectively, and Mβ is the multiplicity of the β irrep being projected. Similar approaches have been employed in other materials47,48.

Group theory

The symmetry-adapted linear combinations of atomic displacements used for the phonon calculations were found using the ISOTROPY software suite64. The subduction analysis to find the connections between irreps through the group-subgroup transition was performed using the CORREL application hosted by the Bilbao crystallographic server46.