An overview of the spin dynamics of antiferromagnetic Mn₅Si₃

Antiferromagnetic (AFM) compounds have recently emerged as a promising class of materials in the field of spintronics.^1,2 They exhibit a plethora of transport properties, such as the anomalous and magnetic spin Hall effects discovered in Mn₃Sn.^3,4 As another example, the unusual tilted spin current discovered in RuO₂⁵ is evidence for the newly recognized class of altermagnetic materials.⁶ These effects are already being exploited in multilayer structures that incorporate either heavy-metal or ferromagnetic layers, for instance to switch the magnetic order parameter using spin–orbit torques at terahertz (THz) frequencies.⁷ 

In fact, one of the attractive aspects of AFM materials for spintronics applications is the much faster magnon dynamics in comparison with ferromagnets.⁸ Magnons can also carry spin currents over long distances, as found in α-Fe₂O₃⁹ and in MnPS₃.¹⁰ In addition, magnons in AFM materials can display topological properties^11,12 that parallel those found for electrons,¹³ as exemplified by the Dirac magnons discovered in Cu₃TeO₆¹⁴ and in CoTiO₃.¹⁵ Therefore, combined theoretical and experimental investigations of magnons in AFM materials are essential to shed light on the fundamental physical effects and on how to adapt them to prospective spintronics devices.

In this article, we focus on the multifunctional AFM Mn₅Si₃. This material exhibits interesting thermodynamic properties, such as the inverse magnetocaloric effect¹⁶ and thermomagnetic irreversibility,¹⁷ and non-trivial transport properties through the anomalous Hall effect.¹⁸ Hence, its macroscopic properties and crystal and magnetic structures have extensively been studied over the years.^16–25 Moreover, a recent classification and description based on spin-symmetry principles proposes that Mn₅Si₃ belongs to a new distinct magnetic class called altermagnetism.⁶ As it consists of cheap and abundant elements and can be grown as a polycrystal,^17,19 single crystal,^24,26 thin film,^18,27 nanoparticle,²⁸ and nanowire,²⁹ it may attract further attention for the design and manufacturing of novel devices based on spintronic technologies. Here, we microscopically probe the magnetization dynamics of Mn₅Si₃ in the meV regime (THz frequencies) utilizing inelastic neutron scattering (INS) measurements and density functional theory (DFT) calculations. This combined approach is ideal for studying the spin-excitation spectrum under external parameters, such as temperature (T) and magnetic field (H).

Bulk Mn₅Si₃ is paramagnetic (PM) at T > 100 K and crystallizes in the hexagonal space group P6₃/mcm with two distinct crystallographic positions for the Mn atoms (sites Mn1 and Mn2).¹⁹ Below the PM critical temperature, the onset of a long-range magnetic order reduces the symmetry to orthorhombic.^20,21 Such a structural change is absent in epitaxially grown thin films.^27,30 In the temperature 66 < T < 100 K, the magnetic structure of Mn₅Si₃ is collinear (AFM2 phase) with only two-thirds of the Mn2 atoms having finite and ordered spins, which are aligned parallel and antiparallel to the b-axis. The Mn1 atoms and the remaining one-third of the Mn2 atoms have vanishing or non-ordered moments^19,20,23 as shown in Fig. 1(b). For even lower temperatures (⁠$<66$ K), the system realizes a noncollinear spin structure (AFM1 phase), where the Mn1 spins are now ordered but one-third of Mn2 spins remains as in the AFM2 phase. The noncollinearity is a result of frustration and anisotropy, where the Mn1 spins align mostly along the easy axis b, while the two-thirds of the Mn2 spins have significant components along the c-axis as illustrated in Fig. 1(a).²⁴ 

Both AFM phases are the so-called mixed magnetic phases³¹ embodying the coexistence of magnetically ordered and non-ordered Mn sites.^19–21 The magnetic phase diagram as a function of temperature and the magnetic field applied along the c-axis²⁵ is shown in Fig. 1(c). Below T = 66 K, the increasing magnetic field results in transitions from the AFM1 phase to another intermediate AFM1′ phase before reaching the AFM2 phase. In the temperature range 66 < T < 100 K, the AFM2 phase remains stable up to the investigated field of 10 T. We also note that apart from the transitions occurring in the H–T phase diagram^17,25 and presented in Fig. 1(c), there are indications of another AFM phase^19,25,32 appearing at small magnetic fields at low temperatures (H < 1 T, T < 30 K) and surviving in a narrow magnetic field range,²⁴ but that is outside the scope of this work.

In our studies, we used the orthorhombic coordinate system and the scattering vector Q is expressed in Cartesian coordinates Q = (Q_h, Q_k, Q_l) given in reciprocal lattice unit (r.l.u.). The wave vector q is related to the momentum transfer through ℏQ = ℏG + ℏq, where G is an AFM zone center and G = (h, k, l). Several aspects of the magnetism of Mn₅Si₃ were addressed in our previous studies by combining INS measurements and DFT calculations.^16,23,24 The ground state spin configuration (AFM1 phase) and the different magnetic structures induced at low temperatures under magnetic field were theoretically determined.²⁴ The spin-wave spectra were studied in detail in the AFM1²⁴ (T = 10 K) and AFM2²³ (T = 80 K) phases both theoretically and experimentally. These zero-field spin dynamics results will be summarized in Sec. IV together with the new results obtained for the spin-wave spectra for the phases reached under a magnetic field (AFM1′ and field-induced AFM2).

In addition, an insight into how the high-energy spin dynamics (above ≈2 meV) evolves between the AFM1 and AFM2 phases was given in Ref. 16. It was shown that the spin dynamics in the AFM2 phase resembles that of the PM state: the response is broad in Q with a single peak centered at the AFM zone center G in contrast to the spin-wave spectrum of the AFM1 phase having two peaks on both sides of G. In Ref. 16, the peculiar spectrum of the AFM2 phase was analyzed as arising from the coexistence of spin waves and spin fluctuations given the fact that the response obtained by polarized INS experiment is distinct between the AFM2 and PM states. This interpretation was made using the simplest possible hypothesis, but it cannot be ruled out that a more complex dynamical response occurs. Therefore, in the present study, we will name the signal that is characteristic of the AFM2 phase above ∼2 meV as “broad fluctuations.” Its physical origin is ascribed to non-magnetic Mn sites occurring within the magnetically ordered phase. The inverse magnetocaloric effect (cooling by adiabatic magnetization) was related to this change of spin dynamics from well-defined spin waves to broad fluctuations under field.¹⁶ 

This paper is organized as follows: after recalling briefly the methods that are similar to previous papers^16,23,24 (Sec. III), an overview of the spin configuration and spin-wave spectra is given for all phases of the H–T phase diagram in Sec. IV A. Then, a focus on the temperature and magnetic field dependence of the spin dynamics is made at the low-energy (Sec. IV B) and high-energy (Sec. IV C) regimes.

A Mn₅Si₃ single crystal (the same used in previous studies^16,23,24) with a mass of about 7 g and grown by the Czochralski method was oriented in the [100]/[010] scattering plane of the orthorhombic symmetry. Unpolarized INS measurements were carried out on the cold triple-axis spectrometers (TAS) IN12³³ and ThALES at the Institut Laue-Langevin (ILL) in Grenoble, France. The instrument configurations are standard and are given in previous papers.^16,23,24 Inelastic scans were performed with constant k_f = 1.5 Å⁻¹, where k_f is the wave vector of the scattered neutron beam. Spin dynamics investigations under temperature and magnetic field were carried out using a 10 T vertical field magnet.

The magnetic parameters were computed with the DFT code JuKKR,^34,35 which are then used to solve a spin Hamiltonian in the linear spin-wave approximation.^36,37 The Monte Carlo Metropolis algorithm as implemented in the Vampire package^38,39 was employed. A macrocell with 1 μm³ was considered with an equilibration time of 10⁵ Monte Carlo steps and statistical averaging over 10⁵ steps.

Figure 2 shows for each AFM phase of Mn₅Si₃ the computed spin structure and the corresponding magnon dispersion relations originating from the purely magnetic Bragg peaks (2,1,0) and (0,3,1). The AFM2 [Fig. 2(a)] and AFM1 [Fig. 2(b)] phases were previously addressed in Refs. 23 and 24, respectively, and in the present paper, we extend the results to the AFM1′ and field-induced AFM2 phases. The DFT calculations²³ for the zero-field AFM2 phase [Fig. 2(a)] reproduce the experimental magnetic structure determined by neutron diffraction.²⁰ As previously mentioned, this phase has the peculiarity that only two-thirds of the Mn2 sites have ordered magnetic moments. At temperatures below 60 K, the Mn1 sites acquire a finite magnetic moment that leads to a destabilization of the collinear AFM2 ordering. In DFT, a finite moment at the Mn1 sites is achieved in a spin polarized calculation.²⁴ At zero field, we have the AFM1 phase [Fig. 2(b)], which is noncollinear due to the frustrated antiferromagnetic coupling between the Mn1 and Mn2 moments. An applied magnetic field can induce phase transitions into the AFM1′ [Fig. 2(c)] and the so-called field-induced AFM2 [Fig. 2(d)] phases, where the moment at the Mn1 sites is still finite.

We point out that there is a degree of uncertainty in the obtained magnetic interactions, which affects the phase boundaries between the different AFM phases shown in Fig. 1(c). Therefore, the quantitative disagreement²⁴ between the experimental and the theoretical phase boundaries in the H–T phase diagram can probably be ascribed to the uncertainties in the parameterization of the spin model. However, the PM critical temperature of the model Hamiltonian is 110 K, as determined from Monte Carlo simulations, which is in good agreement with the measured transition temperature of 100 K.^17,23 This suggests that the obtained exchange interactions from our calculations are in the correct energy scale of the system.

In contrast to Ref. 23 that dealt uniquely with the AFM2 phase, in the modeled spin-wave spectrum, the Hamiltonian parameters are not rescaled to account for thermal fluctuations for the zero-field collinear AFM2 phase observed experimentally at 66 ≤ T ≤ 100 K [see Fig. 1(c)]. In the calculated spin waves, we observe for E ≥ 2 meV similar characteristics in the magnon bands of the AFM1, AFM1′, and field-induced AFM2 phases despite their differences in the magnetic structure. All these phases share the same Hamiltonian (exchange couplings and anisotropy). The difference is that the applied field causes a change in the ground-state spin configuration but only small changes in the high energy magnon spectra. This seems to indicate that the spin configuration does not affect strongly the spin-wave properties as the interactions between spins do.

Another feature that becomes apparent is that the spectrum of the zero-field AFM2 phase consists only of an acoustic mode in the energy range of 2 ≤ E ≤ 25 meV, as can be seen in Fig. 2(a). In contrast, all the other phases feature optic bands originating from the AFM zone centers at about E = 12 meV. This difference can be explained by the vanishing of the magnetic moments in the Mn1 sites in the zero-field AFM2 phase. Thus, the optic mode is mainly due to the AFM coupling between spins in the Mn1 and Mn2 sites (|J| ∼ 2 meV²⁴).

If we now focus on the low-energy spin dynamics (E < 2 meV), we recall that there is a single anisotropy gap of about 0.8 meV in the AFM1 phase²⁴ and a double gap of about 0.2 and 0.4 meV in the AFM2 phase²³ ascribed to the bi-axial anisotropy [see Fig. 3(b)]. We also note that although Mn atoms with non-ordered moments are present in all phases, the effects associated with fluctuating moments are difficult to model theoretically. However, their fingerprint can be detected through neutron scattering experiments (see Secs. IV B and IV C).

Starting from the AFM1 phase, the temperature and magnetic field dependence of the low-energy spin dynamics (E < 2 meV) was not addressed in our previous studies. This is done here through INS measurements around the magnetic zone centers G = (0, 3, 0) and G = (1, 2, 0) (see Fig. 3). Figure 3(a) shows the temperature dependence of the spin excitations obtained at H = 0 T. A first peak is always centered at E = 0 meV and corresponds to the elastic line, while the second one at finite E points to the existence of gapped spin waves. In the constant-Q scans, a sharp rise in the signal at 0.3 < E < 1.2 meV is followed by a large tail extending to high energies [see Figs. 3(a) and 3(c)]. This tail could arise from resolution effects in relation to the steep magnon dispersion in the AFM1 phase.²⁴ Another scenario is that spin fluctuations originating from non-ordered moments would result in a broad quasielastic contribution extending to high energies and being overlaid over the signal of well-defined spin waves.

Within the resolution of our measurements, we observe for the AFM1 phase a single gap that is opening with decreasing temperature [see the red data points in Fig. 3(b)]. This spin gap hints to a local easy axis existing within this noncollinear phase, and our DFT calculations indicate that the easy axis of the AFM1 phase is b. Note that in the AFM2 phase, the primary and secondary easy axes are b and c, respectively,²³ resulting in a double spin gap [see the black data points at T = 80 K in Fig. 3(b)]. The higher energy gap of the AFM2 phase is solely determined by the b-axis anisotropy parameter, while the lower energy gap involves the difference of anisotropy parameters between the b and c-axes.²³ Consistently, one can see in Fig. 3(b) that the temperature dependence of the AFM1 gap extrapolates to the value of the higher energy gap of the AFM2 phase. Naturally, no spin gap is observed in the PM state since spin fluctuations due to short range correlations result in a continuum of states [see the orange data point at T = 120 K in Fig. 3(b)].

By applying a field parallel to the c-axis in the AFM1 phase, we induce phase transitions [see Fig. 1(c)] that also manifest in the low-energy spin dynamics. The spectra obtained at G = (1, 2, 0) at T = 50 K as a function of field are shown in Fig. 3(c). First, weak magnetic fields (H ≤ 2 T) do not affect the gap in the AFM1 phase; however, for higher fields (H ≥ 6 T), the gap increases and splits, recovering a double-gapped structure characteristic of the zero-field AFM2 phase [see T = 80 K in Fig. 3(b)], but here for the field-induced AFM2 phase (T = 50 K and H = 8 and 10 T). The obtained field dependence of the spin gaps at T = 50 K is shown in Fig. 3(d). While the AFM1 structure is complex with components of ordered moments parallel and perpendicular to the applied field, the gap does not change between 0 and 2 T. The value of the gap in the middle of the AFM1′ phase (at H = 4 T) is similar to the one in the AFM1 phase and increases at 6 T in the boundary with the field-induced AFM2 phase. In contrast, a clear variation can be addressed for the field-induced AFM2 phase. Starting from the AFM1 phase at 50 K and applying a magnetic field, the behavior is the same as that for the 80 K AFM2 phase under field for the same field range of 6–10 T: the (zero field) higher energy mode does not depend on the magnetic field, while the (zero field) lower one increases monotonically with field (see Fig. 4 of Ref. 23; note also that the modes cross at 4 T). In this former paper (Ref. 23), the field response of the spin waves in the AFM2 phase at 80 K was theoretically explained by considering the precession directions of the different magnon modes.

Figure 4(a) shows the evolution of the constant energy spectra collected along Q = (Q_h, 2, 0) at 2 meV at T = 50 and 80 K at zero field. It is similar to the data previously obtained at slightly higher energies of 3 and 5 meV¹⁶ and illustrates the change in the spin dynamics from two separate spin-wave peaks (T = 50 K) to broad fluctuations (T = 80 K) as described in Sec. II. A similar change in the spectra can also be obtained at 50 K as a function of field as shown in Fig. 4(b) for 0 and 10 T. Since the temperature is now fixed in this measurement, this indicates that the origin of the fluctuations is not a thermal broadening but that they are characteristic of the AFM2 phase.

The overall dynamical response is also depicted in Fig. 4(c) in a color-coded map of the INS intensity measured for another constant energy transfer of E = 3 meV at T = 50 K as a function of Q = (Q_h, 2, 0) and magnetic field. As shown in Figs. 4(b) and 4(c), at T = 50 K, as the external magnetic field increases within the AFM1 and AFM1′ phases, the two peaks remain resolved and unchanged (a consistent behavior with the calculated spectra shown in Fig. 2), until the field-induced AFM2 phase is reached. Then, the signal is substituted by a broad fluctuation pattern. In the color-coded map [Fig. 4(c)], this manifests through the change in the spectra from two intensity ridges at Q_h ≃ 0.9 r.l.u. and Q_h ≃ 1.1 r.l.u. to one intensity ridge centered at Q_h = 1 r.l.u.

In order to further support our statements, data on the temperature dependence of the spin-excitation spectrum of Mn₅Si₃ at 0 and 10 T are shown in Fig. 5, where the magnetic Bragg peak intensities are used to track the phase boundaries in the H–T diagram. The temperature dependence of the peak intensities at G = (1, 2, 0) and G = (0, 3, 0) at H = 0 T and H = 10 T is shown in Fig. 5(a). At H = 0 T, with decreasing temperature, the abrupt increase in intensity for the (1, 2, 0) Bragg peak first at T = 100 K and then at T = 66 K marks the onset of the AFM ordering, in good agreement with the phase diagram shown in Fig. 1(c). The (0, 3, 0) reflection is absent in the AFM2 phase²¹ and is used for confirming the AFM1 (or AFM1′) ordering. At H = 10 T, the transition temperature of the AFM2 phase, as seen from the (1, 2, 0) peak intensity, is weakly affected. The (0, 3, 0) peak intensity indicates that a phase transition occurs at about 40 K (AFM2 → AFM1′), which is in agreement with a small intensity change observed for (1, 2, 0) at the same temperature. At T < 40 K and H = 10 T, Mn₅Si₃ is still in the AFM1′ phase and magnetic fields higher than 10 T may be needed to reach the field-induced AFM2 phase at low temperatures [we recall that the construction of the phase diagram shown in Fig. 1(c) is based on macroscopic measurements]. Our neutron data corroborate previous results about the field-induced transitions in Mn₅Si₃.^{17,19,22,25,32,40}

The corresponding changes in the inelastic response measured along Q = (Q_h, 2, 0) at 3 meV are shown in Figs. 5(b) and 5(c). At zero field, the switch in the spin dynamics manifests with increasing temperature as previously described from the change of two spin-wave intensity ridges to a single ridge centered at Q_h = 1 r.l.u. that resembles that of the PM state (T > 100 K). At 0 T, this occurs at 66 K (AFM1 → AFM2 transition), while at 10 T, the switch between the AFM1′ and field-induced AFM2 phases occurs around 40 K. Therefore, the broad fluctuations are now observed in a wider temperature range from 45 to 100 K, stressing again that they do not correspond to a thermal broadening of the spin waves of the AFM1 (or AFM1′) phase but that this excitation spectrum is an intrinsic feature of the AFM2 phase. Altogether, the change of the spin dynamics between AFM1 (or AFM1′) and AFM2 is observed along three different paths in the H–T phase diagram: temperature dependence at 0 T [Fig. 5(b)], field dependence at 50 K [Fig. 4(c)], and temperature dependence at 10 T [Fig. 5(c)].

In Mn₅Si₃, the peculiar excitation spectrum of the AFM2 phase is naturally associated with the presence of ordered and non-ordered Mn sites. This is based on the observation that it looks at first sight similar to the PM state, but it also involves spin waves as demonstrated by polarized INS experiments in Ref. 16. This response could be a complex entanglement between spin waves and spin fluctuations beyond the simple sum proposed in Ref. 16, and therefore, in the present paper, we named it broad fluctuations. Whatever its precise nature is, one concludes that the different features of the spin dynamics characteristic of the AFM1 or AFM2 phase are linked to changes of the magnetic behavior of specific Mn sites as a function of temperature and magnetic field.

At present, it is unclear whether the observed fluctuations arise from the Mn1 sites or the one-third non-ordered Mn2 sites. Our theoretical results so far do not reproduce the broad features of the spin-excitation spectra of the zero-field and field-induced AFM2 phases (see the top and bottom panels in Fig. 2) since fluctuating moments (spins changing their sizes and directions) are, at the present stage, hard to characterize from DFT calculations. Our model indicates the presence of magnetic moments on the Mn1 sites in the field-induced AFM2 phase (see Fig. 2 bottom left). However, based on a three-spin model introduced in Ref. 24, it was demonstrated that a non-rigid Mn1 moment in a frustrated environment can fluctuate and its size change depends on the competition of the exchange interactions and the magnetic field strength. Concerning the one-third of Mn2 sites in the field-induced AFM2 phase, a neutron diffraction study proposes that there is no induced moment under an external magnetic field;²² however, neutron scattering is not directly a site selective probe and further hypotheses must be made when discussing such aspects. Last but not least, one needs to consider that the dynamical response of the non-ordered moments may manifest in other (Q, E) regions that were not investigated in our INS study in either the AFM1 (AFM1′) phase or the AFM2 phase.

The dynamics of the fluctuations within the mixed magnetic phases of Mn₅Si₃ could be potentially studied by various experimental techniques probing different energy scales, such as AC susceptibility, neutron spin echo, nuclear magnetic resonance (NMR), and muon-spin rotation (μSR). In particular, local probes, such as NMR⁴¹ and μSR, can be employed to investigate the magnetization dynamics at specific crystallographic sites and obtain information that is not accessible by neutron experiments. Apart from Mn₅Si₃, mixed magnetic phases are found in several metallic and insulating magnetic compounds.^42–48 The possible consequences of the fluctuating or non-ordered moments in the physical properties of interest for spintronics are not yet addressed to our knowledge.

Finally, we mention that besides the impacts of temperature and magnetic field, an increasing amount of evidence points out that pressure could significantly affect the magnetic structure and, consequently, the magnon spectrum of Mn₅Si₃. Magnetization measurements under hydrostatic pressure indicate a destabilization of the AFM1 phase with increasing pressure, leading eventually to ferromagnetic ordering.⁴⁹ The chemical pressure induced by carbon implantation,⁵⁰ by doping the Mn-site^51–53 or the Si-site,^54,55 is demonstrated to have drastic changes on the electric and magnetic properties of Mn₅Si₃. The inherent stress Mn₅Si₃ single crystals acquire when grown by different techniques is proposed to lead to deviations on the H–T phase diagram shown in Fig. 1(c).²⁶ Finally, epitaxial strain may potentially stabilize the AFM2 ordering to a wider temperature range.³⁰ Note that for these very thin films, it is proposed from the measured physical properties that the AFM2 ordering is of different kind than in the bulk material and occurs without a doubling of the unit cell.³⁰ Therefore, a dedicated study on the effects of uniaxial or hydrostatic pressure on the spin structure of bulk Mn₅Si₃ would be a first essential step to elucidate the microscopic mechanisms responsible for the results reported so far and to investigate the response of the ordered and non-ordered Mn sites under this thermodynamic parameter.

In this study, we investigated the evolution of the spin-excitation spectrum of Mn₅Si₃ under external parameters. The changes of the spin structure stimulated by temperature and magnetic field are also traced in the spin-excitation spectrum of Mn₅Si₃. At low energies, all AFM phases host spin-wave excitations in agreement with theoretical calculations. The temperature or field dependence of the spin gap allows a tracking of the changes in the spin-excitation spectrum between the AFM1 and AFM2 phases. Apart from the spin gap (single vs double), the high-energy spin dynamics is also a fingerprint of each magnetic phase. At high energies, whatever the combination of temperature and applied field, the spin dynamics of the AFM1/AFM1′ and AFM2 phases are very robust at consisting of either spin waves or broad fluctuations. The broad fluctuation pattern of the AFM2 phase at this stage cannot be addressed by our theoretical framework where on the magnetic sites (Mn1 and/or two-third of Mn2 sites), rigid moments are assumed. Solving this puzzle likely requires a theory that extends the Heisenberg model to account for low-energy longitudinal spin fluctuations of the nominally non-magnetic Mn sites. While the present overview gives a clear picture of the spin-excitation spectrum of Mn₅Si₃, the precise microscopic relationship between the peculiar spin dynamics and the evolution of the magnetically ordered and non-ordered Mn sites across the H–T phase diagram is still a topic to be experimentally and theoretically addressed.

N.B. acknowledges the support of JCNS through the Tasso Springer Fellowship and the Czech Science Foundation GAČR under the Junior Star Grant No. 21-24965M (MaMBA). F.J.d.S. acknowledges the support of the European H2020 Intersect project (Grant No. 814487), and N.M. acknowledges the support of the Swiss National Science Foundation (SNSF) through its National Centre of Competence in Research (NCCR) MARVEL. The work of M.d.S.D. made use of the computational support by CoSeC, the Computational Science Centre for Research Communities, through CCP9. This work was also supported by the Brazilian funding agency CAPES under Project No. 13703/13-7 and the Deutsche Forschungsgemeinschaft (DFG) through SPP 2137 “Skyrmionics” (Project No. LO 1659/8-1). We gratefully acknowledge the computing time granted through JARA on the supercomputer JURECA⁵⁶ at Forschungszentrum Jülich GmbH and by RWTH Aachen University.

The authors have no conflicts to disclose.

N. Biniskos: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). F. J. dos Santos: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). M. dos Santos Dias: Investigation (equal); Software (equal); Validation (equal); Writing – review & editing (equal). S. Raymond: Investigation (equal); Resources (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). K. Schmalzl: Investigation (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). P. Steffens: Resources (supporting). J. Persson: Resources (supporting). N. Marzari: Funding acquisition (equal); Resources (equal). S. Blügel: Funding acquisition (equal); Resources (equal). S. Lounis: Funding acquisition (equal); Resources (equal). T. Brückel: Funding acquisition (equal); Project administration (supporting); Resources (equal); Supervision (supporting); Validation (supporting); Writing – review & editing (supporting).

The neutron data collected at the ILL are available at Refs. 57 and 58. The theoretical results that support the findings of this study are available from the corresponding author upon reasonable request.