Non-centrosymmetric, transverse structural modulation in SrAl4, and elucidation of its origin in the BaAl4 family of compounds

At ambient conditions SrAl4 adopts the BaAl4 structure type with space group I4/mmm. It undergoes a charge-density-wave (CDW) transition at TCDW = 243 K, followed by a structural transition at TS = 87 K. Temperature-dependent single-crystal X-ray diffraction (SXRD) leads to the observation of incommensurate superlattice reflections at q = \sigma c* with \sigma = 0.1116 at 200 K. The CDW has orthorhombic symmetry with the acentric superspace group F222(00sigma)00s, where F222 is a subgroup of Fmmm as well as of I4/mmm. Atomic displacements mainly represent a transverse wave, with displacements that are 90 deg out of phase between the two diagonal directions of the I-centered unit cell, resulting in a helical wave. Small longitudinal displacements are provided by the second harmonic modulation. The orthorhombic phase realized in SrAl4 is similar to that found in EuAl4. Electronic structure calculations and phonon calculations by density functional theory (DFT) have failed to reveal the mechanism of CDW formation. However, DFT reveals that Al atoms dominate the density of states near the Fermi level, thus, corroborating the SXRD measurements. SrAl4 remains incommensurately modulated at the structural transition, where the symmetry lowers from orthorhombic to b-unique monoclinic. We have identified a simple criterion, that correlates the presence of a phase transition with the interatomic distances. Only those compounds XAl4-xGax(X = Ba, Eu, Sr, Ca; 0

EuAl 4 , containing divalent Eu, is a 3D CDW compound that also undergoes four successive antiferromagnetic (AFM) transitions below 20 K [37][38][39][40][41].It has been reported to have a chiral spin structure and skyrmions [41,42], similar to skyrmions reported in other divalent Eu-based systems, such as EuPtSi [43,44].A skyrmion state of magnetic order, and the non-trivial band topology were furthermore established by measurements of the topological Hall resistivity, the muon-spin rotation and relaxation (µSR), and the magnetostriction [33,35,45].Our recent investigation into the modulated structure of EuAl 4 reveals a breaking of the fourfold rotational symmetry, resulting in an orthorhombic F mmm symmetry of the CDW state below 145 K.There is no evidence of a lattice distortion away from tetragonal symmetry.The reduction towards orthorhombic symmetry is brought about solely by the transverse CDW modulation in EuAl 4 .More recent work proposes the loss of inversion symmetry [46,47].Alternatively, centrosymmetric Immm symmetry was proposed for the CDW state of EuAl 4 on the basis of inelastic x-ray scattering experiments [48].
Replacement of the divalent rare-earth element Eu by divalent alkaline earth metals Ba, Sr and Ca results in isostructural compounds.SrAl 4 and CaAl 4 undergo CDW transitions or other structural transitions (STs).However, BaAl 4 does not undergo any phase transition [32,34].Out of all the materials in the XAl 4 series, the compound that is most similar to EuAl 4 would be SrAl 4 .SrAl 4 exhibits a CDW transition at T CDW = 243 K, which is at a much higher temperature than the CDW transition of EuAl 4 [49,50].Isostructural CaAl 4 undergoes a ST at 443 K, which is at an even higher temperature than for SrAl 4 [51].SrAl 4 undergoes a second transition at T S = 87 K [49].It was suggested that the symmetry of the low-temperature phase is monoclinic [49].The mechanism of CDW formation in either SrAl 4 or EuAl 4 presently is not understood, as both compounds possess a complex 3D electronic structure [37,49,52,53].
Here, we present the results of temperature-dependent single-crystal x-ray diffraction (SXRD) experiments on SrAl 4 .We find that the CDW transition is accompanied by a reduction in symmetry from tetragonal to F -centered orthorhombic, analogous to EuAl 4 [40].
However, second-order satellite reflections in the SXRD data set of SrAl 4 point towards a loss of inversion symmetry, resulting in the superspace group F 222(0 0 σ)00s for the CDW phase.
Reconsideration of our SXRD data on EuAl 4 shows a marginally better fit for F 222(0 0 σ)00s than for originally published F mmm(0 0 σ)s00 [40].The lack of resolving power of those data probably is the result of the lack of second-order satellite reflections in the SXRD data on EuAl 4 [40].Nevertheless, details of both structure models for EuAl 4 are provided in the Supplemental Material [54].
Either model for EuAl 4 and SrAl 4 leads to qualitatively similar variations of interatomic distances along the incommensurate coordinate of the CDW in these compound.This confirms the earlier conclusion, that the network of Al atoms governs the CDW in SrAl 4 as well as in EuAl 4 [40].This finding is supported by solid solution samples of isostructural SrAl 4−x Si x and SrAl 4−x Ge x , since the CDW is suppressed by the disorder at the Al site, and induces superconductivity in the case of Si-doping [55,56].From SXRD and physical properties measurements, we confirm that below 100 K SrAl 4 undergoes a second phase transition that is characterized by a monoclinic lattice distortion, but across which the CDW modulation remains incommensurate.In this paper, we present the modulated crystal structure of SrAl 4 , and we discuss the similarities and subtle differences in the CDWs of SrAl 4 and EuAl 4 .Furthermore, we establish that the value of c/a must be within the narrow range 2.51 < c/a < 2.54, in order for a compound of this family to undergo a phase transition.This allows us to predict whether new materials developed in this series could or could not undergo a CDW or structural transition.

II. EXPERIMENTAL AND COMPUTATIONAL DETAILS
A. Crystal growth Single crystals of SrAl 4 were grown by the Al self-flux method according to [49].Crystals were grown at three places, using similar methods.
At the Lab. of Crystallography in Bayreuth, the elements strontium (Alfa Aesar, 99.95% purity) and aluminium (Alfa Aesar, 99.9995%) were filled into an alumina crucible in the ratio 1:19.The crucible was sealed in an evacuated quartz-glass ampoule.It was heated to a temperature of 1173 K and held at this temperature for 2 hours.The crucible was then cooled to 823 K with a rate of 0.5 K/hour, at which point the crystals were separated from the molten metal by centrifugation.Small crystals were selected and annealed in vacuum for 72 hours at 723 K.The 1:4 stoichiometry of the product was confirmed by structure refinement against SXRD data.The resulting crystals A were used for the SXRD experiments.
At the Tata Institute in Mumbai, the elements were filled into an alumina crucible with the ratio 1:23.Crystal growth was at 1323 K for 24 hours, after which it was cooled to 973 K with a rate of 1 K/hour, followed by centrifugation.The longer growth time resulted in crystal B that is larger than crystal A. These crystals were employed without annealing for measurement of the specific heat.At the Department of Quantum Matter in Hiroshima, the same procedure was followed, resulting in crystal C that was used for the measurement of the electrical resistivity.corresponding to a rotation of the crystal over 364 deg, which was repeated 10 times.These data were binned to a data set of 364 frames of 1 deg rotation and 10 s exposure time [57].
Further details are provided in the Supplemental Material [54].

C. Single-crystal x-ray diffraction data processing
The EVAL15 software suite [58] was used for processing the SXRD data.Each temperature comprises two runs, one with and another without 2θ offset of the detector.The two runs were integrated separately, and subsequently merged in the module ANY of EVAL15.SADABS [59] was used for scaling and absorption correction with Laue symmetry 4/mmm for the data measured at 293 K (periodic phase).Different Laue symmetries were employed for the runs collected at 200, 120 and 100 K (CDW phase), depending on the symmetry of the structure model being tested (compare Tables S2-S4 in the Supplemental Material [54]).
Laue symmetry 2/m (b-unique) was used for the SXRD data at 20 K (low-temperature phase).The resulting reflection files were imported into the software JANA2006 [60,61] for structure refinements.Tables I and II, and Tables S2-S6 in the Supplemental Material [54] give details of the analysis, crystallographic information, atomic coordinates and modulation amplitudes.
The conventional unit cell was fully relaxed with a 12 × 12 × 4 k-mesh sampling until the energy convergence tolerance fell below 10 −7 eV and the force was less than 10 −3 eV/ Å.
The cutoff energy of the plane-wave basis was chosen as 520 eV.The DFT optimized results are close to the experimental values (Table I).An 18 × 18 × 18 k-mesh was implemented for the Brillouin zone (BZ) integral sampling of the primitive unit cell.The Fermi surface was calculated by using the WannierTools package [67] with tight-binding Hamiltonian constructed by the WANNIER90 code [68,69], which is based on the maximally localized Wannier function (MLWF) method [70].The bare charge susceptibility is carried out using a k-mesh of 200 × 200 × 200.The phonon spectrum calculations were performed within the framework of density functional perturbation theory (DFPT) [71,72] and the finite displacement method, implemented by the PHONOPY [73] package combined with VASP, and by Quantum ESPRESSO [74,75].

E. Physical properties
The dc electrical resistivity was measured between 6 and 296 K, employing a standard four-probe method.The small size of the specimen prevented the identification of lattice directions, so that the single experimental run has resulted in ρ(T ) along an unspecified, general direction.Basically, the temperature dependence of the electrical resistivity confirms the data reported in Ref. [49].The present resistivity data are given in Section S6 of the Supplemental Material [54].A commercial physical property measurement system (PPMS, Quantum Design, USA) was used for measuring the specific heat data in a heating run.

III. RESULTS AND DISCUSSION
A. The orthorhombic CDW phase SrAl 4 has been reported to undergo two phase transitions: one at T CDW = 243 K and another below 90 K [49].Cooling the crystal from 293 to 200 K revealed superlattice reflections with q = 0.1116(2) c * at 200 K, similar to the diffraction of isostructural EuAl 4 below 145 K [38,40].The data measured at 200 K (incommensurate CDW phase) could be refined successfully (Table I).Like for EuAl 4 [38,40], we do not observe any lattice distortion within the CDW phase, thus preserving the tetragonal lattice symmetry for the basic structure.
In order to elucidate the modulated CDW structure at 200 K, we have tested different superspace groups as symmetry of the crystal structure (See Tables S2-S4 in the Supplemental Material [54]).The candidate models are based on symmetries I4/mmm (model A) and its centrosymmetric (Table S2) and noncentrosymmetric (Table S3) subgroups.The best fit to the SXRD data is obtained for model D with the noncentrosymmetric superspace group F 222(0 0 σ)00s.This superspace group is a proper subgroup of centrosymmetric F mmm(0 0 σ)s00, the latter which was previously found for the CDW phase of EuAl 4 [40].
For SrAl 4 , model C with superspace group F mmm(0 0 σ)s00 also leads to a good fit to the SXRD data, except for the second-order satellite reflections, the latter which have been observed by SXRD, but which are not present in the available data sets for EuAl 4 [40,48,76].
Based on their significantly worse fit to the SXRD data, I-centered orthorhombic and monoclinic superspace groups can be excluded as symmetry for SrAl4 4 .Upon further cooling below 200 K, the length of the modulation wave vector gradually shrinks [Fig.1(d)].As a consequence, main reflections and nearby satellite reflections are not always resolved at 120 K and below (compare the streaky maxima in Fig. 2).A reliable

B. Location of the CDW
SrAl 4 is isostructural to EuAl 4 with a tetragonal crystal structure at room temperature [78], that is preserved as basic structure at lower temperatures (Fig. 3).
The modulation functions exhibit the following features.Firstly, for the centrosymmetric model C, the largest modulation is along a F .It has about twice the amplitude in SrAl 4 , and values are nearly equal for all three independent atoms-Sr, Al1 and Al2-like it is the case for EuAl 4 (Table S6 and [40]).A smaller amplitude is found along b F exclusively for the Al1 atom.Major difference with respect to EuAl 4 is the presence of second-order harmonics with non-zero amplitudes exclusively along c F .Although these values are much smaller than the principal modulation, they might be responsible for the displacements along c suggested by Korshunov et al. on the basis of inelastic x-ray scattering experiments [48].The present results show that the alternative symmetry of Immm(0 0 σ)s00 [48] is not required for achieving z-displacements.S13 in the supporting information.
The situation is slightly different for the noncentrosymmetric model D. The independent atom Al1 splits into two sites, Al1a and Al1b, in such a way that Al1a is bonded to four Al1b and vice versa [Fig.3(b)].Modulations of similar magnitudes are again present along a F for all four independent atoms.However, all atoms now possess modulations along b F , which are of different magnitudes, especially Al1a and Al1b (Table II).The second-order harmonic amplitudes remain small and represent the z displacement.
Surprisingly, the different models C and D as well as the corresponding models for EuAl 4 provide a qualitatively similar picture for the modulations of the interatomic distances.
Largest variation of the bonding contacts is between Al1a and Al1b, with Al2-Al1a and Al2-Al1b of secondary importance (Fig. 4).t-Plots are given in the Supplementary Material [54] for other interatomic distances as well as for the comparison of models C and D, and for a similar comparison for EuAl 4 .We conclude that the Al1 atoms govern the formation of the CDW in both SrAl Only four compounds undergo phase transitions.

C. Criteria for the formation of CDW in BaAl 4 type compounds
Only a few of the isostructural compounds XAl 4−x Ga x (X = Ba, Eu, Sr, Ca; 0 < x < 4) undergo phase transitions.First, it is noticed that chemical disorder tends to suppress a CDW transition, if the disorder is at sites carrying the CDW.The transition temperature is lowered upon increasing doping, while a few percent of doping can be sufficient to completely suppress the CDW transition [81,82].The sensitivity of the CDW for chemical disorder explains the absence of CDW transitions in EuAl 4−x Ga x (0 < x < 4, except x = 0 and x = 2) [31], and the absence of CDW transitions in SrAl 4−x Si x and SrAl 4−x Ge x [55,56].
Thus, restricting the analysis to compounds XAl 4−x Ga x with x = 0, 2 or 4, we make the observation that structural and CDW phase transitions take place in compounds that have the ratio c/a within a narrow range: 2.51 < c/a < 2.54 (Fig. 5).EuGa 4 has a CDW phase transition for pressures exceeding 2 GPa [37].One might then speculate, that the ratio c/a of EuGa 4 at high pressures will be within the critical range (Fig. 5).This criterion is useful to predict CDWs in new compounds of this family.At present there is no information regarding whether SrAl 2 Ga 2 exhibits CDW or other structural transitions.

D. Evidence of monoclinic distortion below 100 K
The orthorhombic phase is realized through the orthorhombic symmetry of the CDW modulation displacements (Table II), while the lattice and basic structure remain tetragonal (Fig. 1 and Table S4).The structural transition at T S = 87 K is accompanied by a lattice distortion.In particular, we find the angle β to be different from 90 deg within the lowtemperature phase [Fig.1(b)].These values indicate that the low-temperature phase is b-unique monoclinic, in agreement with Nakamura et al. [49].
The structural phase transition leads to a twinned crystal, such that the monoclinic lattice distortion is visible as split reflections in q-scans along the direction of c * .For example, for the reflection (0 2 4), a secondary peak appears below 100 K between the main reflection and first-order satellite (Fig. S1 in the Supplemental Material [54]).Furthermore, these q-scans show that the incommensurate CDW satellite reflections persist into the low-temperature phase.Across T S , they continue to grow in intensity and the length of q continues to decrease [Fig.
Due to overlap between main and satellite reflections, as it is the result of the short modulation wave vector [Fig.1(d)], structure refinements failed.However, measured intensities are of sufficient quality to distinguish between tetragonal, orthorhombic and monoclinic symmetries on the basis of R int values for averaging equivalent reflections.These quantities clearly favor monoclinic, b-unique symmetry for the low-temperature phase (Table III).Fermi level, which is not easy to find experimentally.
Using the software IRVSP [83], we found that these two bands belong to the irreducible representations of point group 4mm, marked LD6 (green line) and LD7(orange line), respectively [Fig.6(b)].This indicates that the band crossing is a topologically protected Dirac point.It follows that SrAl 4 is a Dirac nodal line semimetal without SOC and becomes a Dirac semimetal when the SOC is considered, like EuAl 4 [37,40].
The total and atom-projected density of states (DOS) near E F are given in Fig. 6(c).
They reveal that Al states (red line) dominate the DOS near the Fermi level.This observation parallels the situation observed for EuAl 4 [40].Consequently, it is plausible to infer that the CDW is associated with the Al atoms rather than the Sr atoms.
Additional information on the mechanism of CDW formation in SrAl 4 might be obtained from the phonon dispersion relations and the Fermi surface.This result is further checked by the application of alternative computational methods.
The phonon dispersion relations were computed with a highly optimised primitive cell by the finite displacement method.Small imaginary frequencies were obtained on the line M-Γ, when a 3×3×3 k-mesh was used for the computation (see Fig. S18(a) in the Supplementary Material [54]).These imaginary frequencies are eliminated, when the number of k-points is increased toward to 5 × 5 × 5 (Fig. S18(b) in [54]).The software Quantum Espresso [74,75] leads where f (ε n,k ) is the Fermi-Dirac distribution function.The real and imaginary components of the bare charge susceptibility of the SrAl 4 on the k z = 0.11 c * plane are depicted as a function of wave vector q in Fig. S19(a,c) in the Supplemental Material [54].Both components exhibit peak values at the center.We further conducted additional calculations of the bare charge susceptibility along the k z line with k x = k y =0, and present the outcomes in Fig. S19(b,d).The real component shows a peak near the experimentally observed q vector, whereas the imaginary part does not.This discrepancy indicates that Fermi surface nesting (FSN) is insufficient to induce the charge density wave (CDW) in SrAl 4 .These conclusions are consistent with the analysis previously done for SrAl 4 and EuAl 4 [37,40,49], concluding that a clear mechanism of the CDW cannot be determined directly.

F. Specific heat
The temperature dependence of the specific heat (C p ) shows a broad maximum at T CDW of magnitude ∆C p ≈ 12 J mol −1 K −1 (Fig. 7), which is more pronounced than observed for EuAl 4 [40].The measured C p (T ) data do not show any anomaly at the temperature T S of the structural phase transition (Fig. 7).This might be related to the relaxation method of measurement, as employed in the PPMS instrument [85].Alternatively, the absence of a transition at T S might be the result of the presence of lattice defects in as-grown material, as previously described for CuV 2 S 4 [8].

IV. CONCLUSIONS
From SXRD experiments and the physical properties measurements we confirm that SrAl 4 undergoes two phase transitions.At T CDW = 243 K a CDW transition takes place, at which the crystal symmetry is lowered from centrosymmetric tetragonal to noncentrosymmetric orthorhombic.The CDW involves a transverse modulation, while a small longitudinal component exists as second harmonic modulation, i.e. as secondary modulation.The variation of interatomic distances along the incommensurate coordinate demonstrate that Al1 atoms govern the CDW modulation.For the structural transition at T S = 87 K, SXRD unambiguously shows that the low-temperature phase is b-axis monoclinic, and that the CDW is virtually unaffected by this transition.
Replacement of non-magnetic Sr by magnetic Eu has a minimal role in the crystal structure, but it does affect the value of T CDW , which can be explained by different atomic radii of Sr and Eu.This idea has been expanded towards CaAl 4 and BaAl 4 .More generally, isostructural compounds with the tetragonal BaAl 4 structure type exist for XAl 4−x Ga x (X = Ba, Eu, Sr, Ca; 0 < x < 4).We could show that only those compounds undergo phase transitions for which c/a falls within the narrow range 2.51 < c/a < 2.54 (Fig. 5), while CDW transitions may be found only in case of the lack of chemical disorder, i.e.only in compounds with x = 0, 2, or 4.
Both SXRD and electronic band structure calculations support the interpretation that the network of Al atoms is the key contribution to CDW formation, much like in EuAl 4 .
The system possibly possesses a non-trivial band topology and a complex Fermi surface, preventing the mechanism of CDW formation to be simply uncovered in SrAl 4 .
We surmise that it may be possible to realise multiple phases in EuAl 4 , as the latter is proposed to go towards monoclinic symmetry in the AFM phases [40].For CaAl 4 there may exist an intermediary orthorhombic phase between its tetragonal and monoclinic phases.

B
. Single-crystal x-ray diffraction data collection SXRD experiments were performed at Beamline P24 of PETRA III at DESY in Hamburg, employing a four-circle Huber diffractometer with Euler geometry, and radiation of wavelength 0.50000 Å.The temperature of the crystal was controlled by a CRYOCOOL open-flow helium cryostat.Complete data sets were measured at temperatures of 293, 200, 120, 100, 75 and 20 K, covering all phases.Each run of data collection comprises 3640 frames,

FIG. 2 .
FIG. 2. Excerpts of the reciprocal layers of diffraction for (a)-(d) (0 k l) and (e)-(f) (h 0 l), as reconstructed from the measured SXRD data [77].(a)-(d) and (e)-(f) display scattering for temperatures of 293, 200, 100 and 20 K, as indicated.Indices are given for selected main reflections.Selected satellite reflections are indicated by arrows.Up to 3 rd -order satellites are visible at 20 and 100 K. Dark gaps are due to insensitive pixels between the active modules of the PILATUS3X CdTe 1M detector.

FIG. 4 .
FIG. 4. t-Plot of interatomic distances ( Å) in the CDW phase of SrAl 4 at 200 K.(a) d[Al1a-Al1b], and (b) d[Al2-Al1a], d[Al2-Al1b] and d[Al2-Al2], where the first atom is the central atom.t-Plotsdisplay interatomic distances as a function of the phase t of the modulation wave[80].In case of multiple curves, each value of t gives the distances from a central atom towards its neighboring atoms.The number on each curve is the number of the symmetry operator that is applied to the second atom of the bond pair.Symmetry operators are listed in TableS13in the supporting

13 FIG. 5 .
FIG. 5. Value of T c = T S or T CDW as a function of the ratio c/a for eleven compounds XAl 4−x Ga x .

FIG. 6 .
FIG. 6.The bulk band structures of the tetragonal crystal structure I4/mmm of SrAl 4 .(a) Electronic band structure along high-symmetry directions, as obtained without SOC.(b) Electronic band structure calculated with SOC.The dashed lines indicate the Fermi energy, E F .All band crossings are gapped out by SOC, except the crossing of two bands along the M-Γ direction, labeled LD6 (green) and LD7 (orange).(c) The density-of-states (DOS) of the primitive unit cell with SOC.The green line gives the total DOS; the red line gives the contributions of 2s and 2p orbitals of Al; and the blue line stands for contributions of the 3d orbital of Sr d.(d) Phonon dispersion relations obtained by DFPT calculations for a 4 × 4 × 2 supercell of the conventional unit cell, containing 320 atoms.(e) First Brillouin zone of the primitive unit cell.(f) Hole pockets, and (g) electron pockets of the Fermi surface.

Figure 6 (
d) shows the phonon dispersion relations, as computed with aid of a 4 × 4 × 2 supercell of the conventional basic cell, containing 320 atoms.Density functional perturbative theory (DFPT)[71,72] was used with a 4 × 4 × 3 k-mesh.No imaginary frequencies or soft modes can be identified, indicating that electron-phonon coupling (EPC) is too weak to induce a CDW transition.
to the same conclusion [compare Figs.S18(b),(c) and Fig. 6(d)].The Fermi surface of SrAl 4 comprises hole pockets centered on the point M [Fig.6(f)], as well as electron pockets surrounding Γ and those centered on point P [Fig.6(g)].To further check whether Fermi surface nesting (FSN) could responsible for this CDW, the bare charge susceptibility [84] was calculated, employing a k-mesh of 200 × 200 × 200, and considering four conduction bands and four valence bands near the Fermi level, lim ω→0 χ ′′ 0

FIG. 7 .
FIG. 7. Temperature dependence of the specific heat C p of SrAl 4 for 3-280 K, as obtained during heating of the sample.A broad maximum is observed at T CDW = 243 K (indicated by a vertical dashed line).No anomaly could be detected near the second structure transition at T S = 87 K.

TABLE I .
Crystallographic data of crystal A of SrAl 4 at 293 K (periodic phase) and 200 K (CDW phase).Details are given for two different structure models for the CDW phase.

TABLE II .
Amplitudes of the modulation functions of crystal A at 200 K for superspace group F 222(0 0 σ)00s.The relative coordinates, x, y and z of the basic position are also specified.Similar information for F mmm(0 0 σ)s00 is provided in TablesS5 and S6.Values of modulation amplitudes have been multiplied by the corresponding lattice parameter, in order to obtain values in Å.

TABLE III .
Comparison of models on basis of SSG and R int at 20 K. Criterion of observability: