Origin of Charge Density Wave in Topological Semimetals SrAl4 and EuAl4

Topological semimetals in BaAl4-type structure show many interesting behaviors, such as charge density wave (CDW) in SrAl4 and EuAl4, but not the isostructural and isovalent BaAl4, SrGa4 and BaGa4. Here using Wannier functions based on density functional theory, we calculate the susceptibility functions with millions of k-points to reach the small q-vector and study the origin and driving force behind the CDW. Our comparative study reveals that the origin of the CDW in SrAl4 and EuAl4 is the strong electron-phonon coupling interaction for the transverse acoustic mode at small q-vector along the \{Gamma}-Z direction besides the maximum of the real part of the susceptibility function from the nested Fermi surfaces of the Dirac-like bands, which explains well the absence of CDW in the other closely related compounds in a good agreement with experiment. We also connect the different CDW behaviors in the Al compounds to the macroscopic elastic properties.


I. Introduction
The interplay between charge density wave (CDW) and topological phases has received much attention due to the perspective of coupling many-body interaction with non-trivial band structure topology.For example, the nesting Weyl points gapped by chiral symmetry breaking have been proposed to give arise to topological superconductivity 1,2 or CDW with the latter to realize axion electrodynamics 3 for the case of opposite chirality or a topological phase with monopole harmonic order 4 for the same chirality.The Fermi surface nesting (FSN) among Weyl points has been recently studied for the CDW in the quasi-1D (TaSe4)2I 5,6 and the long-wavelength helical magnetic order in NdAlSi 7 .Very recently CDW has been observed in the chiral CoSi 8,9 with Kramer-Weyl points.CDW has also been used 10 to engineer topological phases to realize Dirac points close to the Fermi energy (EF).However, besides FSN, the emergence of CDW also need the important ingredient of electron-phonon coupling 11,12 (EPC).The traditional view of CDW comes from Peierls' instability 13 with perfect FSN in 1D to induce Kohn anomaly or soft phonon mode, which results in a structural transition at low temperature.This ideal 1D picture of CDW has been pervasive in condensed matter physics, but was recently substantially revised for real materials 14,15 .Density functional theory 16,17 (DFT) calculations 18,19 and experiments 20,21 on NbSe2 and TaSe2 have shown that the CDW in these compounds is determined by the real part of linear response or susceptibility function, Re(), with a large EPC interaction to soften the acoustic phonon mode, while not by the FSN from the imaginary part of susceptibility function, Im () , which gives a nesting vector not necessarily corresponding to the CDW q-vector in these specific compounds.
Here by studying the topological semimetals SrAl4, EuAl4 and the related compounds, we take a closer look at the microscopic mechanism for the CDW with topological non-trivial bands and find the critical role of EPC interaction or matrix element play for the CDW in SrAl4 and EuAl4 beyond just FSN of the nodal line Dirac-like bands.
Topological semimetal compounds in the BaAl4-type structure 22 have shown a range of interesting behaviors with a martensitic tetragonal-to-monoclinic structural transition in CaAl4, 23 CDW in SrAl4 and EuAl4, [24][25][26][27][28] and multiple magnetic transitions, topological Hall effects and Skyrmions in EuAl4 and related magnetic compounds at different temperature 25,[29][30][31][32][33][34][35] .The CDW in SrAl4 and EuAl4 has been measured to have a small q-vector along the -Z direction.But there is still a lack of understanding of the origin and driving force behind this CDW.Even more interesting is among the non-magnetic isostructural and isovalent series of SrAl4, BaAl4, SrGa4 and BaGa4, a recent experiment 24 has found that only SrAl4 has CDW at 243 K.Such distinctly different behaviors regarding CDW make these topological semimetals the perfect subjects to test the proposed mechanism 14,15 for CDW as being dominated by EPC interaction instead of FSN.
In this paper, we report a comparative computational study to reveal the origin of the CDW observed in SrAl4 and EuAl4, but its absence in BaAl4 and the two Ga compounds.
We calculate both Re() and Im() using Wannier functions with millions of k-points to reach the small q-vector.Although all of them are topological semimetals with nodal lines in the absence of spin-orbit coupling (SOC) and host Dirac points (DPs) with SOC as protected by the 4-fold rotational symmetry, these DPs are ~0.2 eV above the Fermi energy (EF).The dominating features at EF instead are the Fermi surface (FS) shells formed by the tip and dip of the nodal line Dirac-like band dispersion with two nearby crossings of the EF.With the imperfect FSN from (), the FS shells give the maximum of the real part of susceptibility function, Re &'( , along the -Z direction with a small finite q-vector for all the three Al compounds, but not the Ga compounds.Then among SrAl4, BaAl4 and EuAl4, the EPC calculations from density functional perturbation theory 36 (DFPT) show a larger EPC interaction or matrix element in SrAl4 and EuAl4 than BaAl4 for the transverse acoustic (TA) mode at about the same q-vector of Re &'( along the -Z direction to induce the TA mode to be imaginary.Our study reveals that the origin of the CDW in SrAl4 and EuAl4 is the strong EPC interaction for the TA mode at small q-vector along the -Z direction, besides the Re &'( from the nested FS shells of nodal line Dirac-like bands, which explains well the absence of CDW in the other closely related isostructural and isovalent compounds, in a good agreement with experiment 24 .The different behaviors of the TA mode CDW can also be better understood in connection to the different macroscopic shear modulus and Poisson ratio.We find that the electron charge density redistribution between the Al network and different cation layers indicates the more ionic interaction in BaAl4 than SrAl4 and EuAl4, thus can explain these different CDW and elastic behaviors.It is interesting to find the link between the TA mode CDW with microscopic EPC interaction and the macroscopic elastic properties.

II-a. Topological band structures
Using SrAl4 as an example, the body-centered tetragonal (Pearson symbol tI10) crystal structure in space group 139 (I4/mmm) is shown in Fig. 1(a).The bulk first Brillouin zone (BZ) and (001) surface BZ with the high symmetry k-points are displayed in Fig. 1(b).
The DFT-calculated band structure of SrAl4 is plotted in Fig. 1(c).The highest valence and lowest conduction bands according to simple filling are highlighted in blue and red, respectively.The multiple sawtooth Dirac-like dispersion of the valence and conduction bands near the EF point to a topological semimetal with nodal lines.Indeed, as plotted in Fig. 1(e) for the band gap contours on the (110) plane without SOC, there are a nodal loop around the Z point and also nodal lines away from the -Z direction toward the X point.More nodal lines and loops are plotted on (001) and (100) planes in Supplementary Figure 1.These nodal lines and Dirac-like dispersion are protected by the different sets of mirror symmetries, which are similar to the topological semimetal series of ZrSiS and the related compounds with square lattices of p-orbitals 37 .With SOC, these crossings in nodal lines are all gapped out except for the one along the -Z direction as zoomed in Fig. 1(d).
The point group symmetry is D4h.We have used Vasp2trace 38,39 to analyze the elementary band representations.From the double group representation of D4h for the spinful system, the top valence band along the -Z direction with the 4-fold rotation switches between two different 2-dimensioinal irreducible representations of Λ * and Λ + , thus there is no mixing guarantied at the band crossing, or equivalently speaking, the pair of DPs is protected by the 4-fold rotational symmetry [40][41][42][43][44][45] .Our results also agree with the previous study 22 on BaAl4 for the DPs along the -Z direction.
The locations of the DPs are also indicated in the BZ by the red dots in Fig. 1(b) at the momentum energy of (0, 0, ±0.1912 Å -1 ; EF+0.2186 eV).When projected on the (001) surface, the pair of DPs overlap, but there are still topological surface states converging to the DP projection as shown in Fig. 1(f) along the  , - , direction.The spin texture in Fig. 1(g) shows that these surface states are spin-momentum locked as expected for topological nontrivial surface states.Furthermore, we have also plotted these topological surface states in the other direction of  , - / in Fig. 1(h) with quite different band dispersion, but still they converge to the same DP projection.Because the DPs are 0.22 eV above the EF, the physical properties such as CDW is not directly determined by the DPs, but rather the 3D FS of the nodal line Dirac-like band dispersion near the EF.

II-b. Susceptibility functions and phonon dispersions
The 3D FS of SrAl4 is plotted in Fig. 2 respectively.To study FSN and CDW from band structure, the bare susceptibility function 14 , (), based on DFT single-particle Kohn-Sham bands needs to be calculated in real and imaginary parts, where 6 8, ; and 6 8, ; are the Fermi-Dirac distribution and Delta functions, respectively, of  8, the n-th band energy eigenvalue at the  point with the EF set to zero.
The calculation of () requires very dense double meshes (k and q).Using the maximally localized Wannier functions 46,47 (MLWF), we are able to calculate the 3D susceptibility function () efficiently with millions of k-points in the BZ.
The real part of the 3D susceptibility function Re()/Re &'( of SrAl4 is shown in Fig. 2(d) for the whole BZ with its maximum appearing as two small red spheres along the -Z direction with a small finite q-vector.As plotted along the -Z direction in 1D in Fig. 2(g), the calculated q-vector for Re &'( is 0.23(/c) for SrAl4, agreeing well with the experimental data of 0.22(/c) (dashed line) for the observed incommensurate CDW at 243 K 24 .Such a CDW q-vector has usually been associated with FSN, i.e., the sharp peaks in the imaginary part of susceptibility function, Im(), especially for 1D cases.But as plotted in Fig. 2(e) along the -Z direction, the peak in the Im() of SrAl4 around 0.23(/c) is not sharp, indicating an imperfect nesting due to the 3D nature of the FS in real materials.
The imperfect nesting vector along the kz direction in SrAl4 arises from the pyramid-shape valence FS shell in Fig. 2(b) and is also indicated by an orange arrow in the 2D FS contour on the (110) plane in Fig. 2(f).The calculated phonon band dispersion of SrAl4 is plotted in Fig. 2(h) showing an imaginary TA phonon mode (black arrow) along the -Z direction with a q-vector of 0.24(/c), which matches that of calculated Re &'( .This match of the q-vector of the Re &'( and imaginary TA mode in SrAl4 shows that Re &'( and the imperfect FSN can be an indicator for CDW.However, we will later show that imperfect FSN and the Re &'( does not necessarily guarantee a CDW in the case of BaAl4, because BaAl4 lacks a strong enough EPC to induce a soft phonon mode.
For the other three non-magnetic isostructural and isovalent compounds of BaAl4, SrGa4 and BaGa4, their band structures and FS are plotted in Fig. 3 for comparison to SrAl4.
BaAl4 has a very similar band structure and FS to SrAl4, as shown in Fig. 3(a) and (d), only with small difference at the BZ boundaries.BaAl4 retains the pyramid-shape valence FS shell around the Z point and the interlocked pattern of the hole and electron FS shells.The Im() of BaAl4 as plotted in Fig. 2(e), also has a maximum along the -Z direction, which has a larger q-vector and with a more extended plateau than that of SrAl4.In contrast, the Re &'( of BaAl4 in Fig. 2(g) is at a smaller q-vector than that of SrAl4.This shows a more imperfect FS nesting in BaAl4 than SrAl4 and the mismatch of the peaks in Im() and Re() is not unexpected, because the latter include contributions from the bands away from the EF.As plotted in Supplementary Figure 2, the phonon band dispersion of BaAl4 does not have an imaginary phonon mode or CDW near the Re &'( q-vector.
Next for SrGa4 in Fig. 3

II-c. Electron-phonon coupling and CDW
Despite the similar band structure, FS and also the finite small q-vector for the with the decreasing smearing size (mimicking the electronic temperature) due to the stronger EPC interaction.At the smearing of 0.02 Ry, the TA mode becomes imaginary at a q-vector around 0.24(/c), in a good agreement with the experimentally observed qCDW 24 .
When Sr is replaced by Ba going from SrAl4 to BaAl4 and with the similar FSN, the conventional wisdom is that with a larger mass of Ba than Sr giving lower phonon frequencies, the phonon softening to imaginary mode should be easier and CDW should also exist in BaAl4, assuming EPC interaction stay almost the same.However, both these assumptions are not true for the TA mode along the -Z direction here.Firstly, comparing the phonon dispersion of SrAl4 (Fig.  for both large and small q-vectors.Note the lifetime  NO accounting for the EPC interaction has no mass dependence through  NT by definition.It evaluates the deformation of potential experienced by the electrons on FS due to the structural distortion from phonon modes as convoluted by the Im().The same EPC matrix elements of the deformed potential are also used to calculate the real part of the Green's function for the phonon softening via convolution with Re().In Fig. 4(a), lifetime  NO shows that the EPC interaction is two to three times smaller in BaAl4 than SrAl4 for the TA mode along the -Z direction.Thus, when going from SrAl4 to BaAl4, both the increase in the single-particle (bare) phonon frequency and the decrease in EPC interaction of the TA mode work against the phonon softening in BaAl4.As the results, unlike SrAl4, there is no CDW in BaAl4.
The above analysis focusing on EPC interaction besides imperfect FSN can also explain the behavior of EuAl4, which has a CDW at ~140 K above the Neel temperature, albeit with an even larger mass of Eu than Ba.The band structure and FS of the nonmagnetic EuAl4 (no Eu 4f) in Fig. 5(a) and (b) resemble those of SrAl4 in Fig. 1(c) and 2(a), respectively.The DP has the momentum-energy at (0, 0, ±0.1928 Å -1 ; EF+0.1508 eV).The FS shell of the EuAl4 conduction band is similar to that of SrAl4, except for the larger electron pocket at the N point than SrAl4.The FS shell of the EuAl4 valence band is also similar to that of SrAl4, giving a similar imperfect FSN in () as plotted in Fig. 5(d) and (e) with a slightly smaller nesting q-vector at 0.19(/c) than SrAl4.Noticeably, the Re &'( of EuAl4 has a relatively narrow peak similar to SrAl4 at the small q-vector, rather than the extended plateau of BaAl4.The 3D Re()/Re &'( of EuAl4 plotted in the full first BZ in Fig. 5(c) shows the maximum is indeed only along the -Z direction.For the TA mode dispersion plotted in Fig. 5(e) in comparison to SrAl4 and BaAl4, although in term of atomic mass Sr < Ba < Eu, the TA mode of BaAl4 is the highest among the three in the order of SrAl4 ~ EuAl4 < BaAl4 at the Z point.Then although they all have similar Re &'( q-vector along the -Z direction in Fig. 5(f), the three EPC interactions of the TA mode are quite different.As plotted in Fig. 5(g), the  NO of EuAl4 has an interesting behavior as a function of q when compared to SrAl4 and BaAl4.At small-q (<0.2), the  NO of EuAl4 increases fast like SrAl4, but then plateau like BaAl4 and next decreases at large-q to be even smaller than BaAl4.Although the maximum of  NO for the TA mode in EuAl4 is smaller than that of BaAl4, its value at small-q (<0.2) instead is larger than BaAl4 and closer to that of SrAl4.
The small q-vector range of the EPC interaction is the most relevant to CDW because of the small Re &'( q-vector.Thus, the EPC strength shown by  NO at small-q (<0.2) is in the order of SrAl4 > EuAl4 > BaAl4, explaining why both SrAl4 and EuAl4 have CDW, while BaAl4 does not, even though all three have imperfect FSN and Re &'( q-vector. The TA mode CDW here involves a local shear distortion perpendicular to the caxis.It is very interesting to notice that among the calculated bulk elastic properties (see Table .1),the bulk modulus (B) of 50.6 GPa for BaAl4 is slightly smaller than the 52.9 GPa for SrAl4, reflecting a larger cation size of Ba than Sr, giving both larger a and c lattice constants with a slightly smaller c/a ratio.However, the shear modulus (G) of 36.8GPa for BaAl4 is larger than the 29.1 GPa for SrAl4.This corresponds to a much smaller Poisson ratio of 0.207 for BaAl4 than the 0.268 for SrAl4, which means a compression along the caxis has a less in-plane expansion in response for BaAl4 than SrAl4, i.e., the in-plane interaction is stiffer for BaAl4 than SrAl4.Then moving to EuAl4 with the smallest lattice constants among the three, both B of 57.1 GPa and G of 33.9 GPa increase comparing to SrAl4.But the G of EuAl4 is still smaller than that of BaAl4, resulting in a Poisson ratio of 0.252 similar to that of SrAl4, not BaAl4.Thus, the in-plane interaction in EuAl4 is still softer than BaAl4.
To better understand these differences from electronic structure, we have plotted and compared the electron charge density difference of ( \ ) − () − ( \ ) for X=Sr, Ba and Eu, respectively in Fig. 5 (h-j).The charge density redistributions between the Al network and the different cation layers show that there is more electron transferred from the Ba layer to Al network and also more charge accumulation (yellow) at the boundary between the Ba and Al network than the cases of Sr and Eu.The more ionic character of the interaction in BaAl4 with more in-plane charge accumulation makes it harder for the in-plane shear distortion between the Al network and Ba layer, which explains a much smaller Poisson ratio for BaAl4 than SrAl4 and EuAl4.This also means the TA mode softening for CDW with the local shear distortion in BaAl4 is much harder than that in SrAl4 and EuAl4.It is interesting to find the connection between the CDW with microscopic EPC interaction and the macroscopic elastic properties.

III. Conclusions
In conclusion, using the approaches based on density functional theory, we have , along the -Z direction with a small q-vector for the Al compounds, while it is absence in the Ga ones.Then among the three Al compounds, the electron-phonon coupling (EPC) calculations show a large EPC interaction at small qvector for the transverse acoustic (TA) mode along the -Z direction for SrAl4 and EuAl4, which provides the driving force for the CDW to soften the TA mode, while it is absence in BaAl4.Our study reveals that the origin of the CDW in SrAl4 and EuAl4 is the strong EPC interaction for the TA mode along the -Z direction at small q-vector, besides the Re &'( from the nested FS shells of nodal line Dirac-like bands, which explains well the experimental observations.We also connect the different TA mode CDW distortion to the different macroscopic shear modulus and Poisson ratio.We find that the electron charge density redistribution between the Al network and different cation layers indicates the more ionic interaction in BaAl4 than SrAl4 and EuAl4, thus explains these different CDW and elastic behaviors.

IV. Methods
Density functional theory 16,17 (DFT) calculations have been performed with PBE 48 exchange-correlation functional including SOC using a plane-wave basis set and projector augmented wave method 49 , as implemented in the Vienna Ab-initio Simulation Package 50,51 (VASP).We use a kinetic energy cutoff of 300 eV, Γ-centered Monkhorst-Pack 52 (a).The valence band FS with hole pockets (yellow outside and blue inside) are centered around the Z point, while the conduction band FS with electron pockets (purple outside and green inside) are around the  point.Because of the Dirac-like dispersion, when the bands cross the EF twice nearby along the same high symmetry directions with a dip or tip of the Dirac-like bands, they form FS of thin shells instead of space-filled objects.Such features can be clearly seen when the 3D FS are separated into the hole and electron pieces with a near top-view in Fig.2(b) and (c), (b) and (e), besides the extra band inversion at the S point, the main difference is the upshift of the valence bands along the -Z direction.The inner shell of the hole pockets at the Z point almost merge into a single large pocket with the next valence band touching the EF along the Z-S1 direction to form a small ring-shape pocket (red) around the Z point.As the results, the Re &'( q-vector along the -Z direction for SrGa4 is reduced toward q=0 as plotted in Fig.2(g).SrGa4 does not have a CDW either as shown by the absence of imaginary phonon dispersion in Supplementary Figure2.Moving to BaGa4 in Fig.3(c), the top valence band along the -Z direction is pushed to even higher energy with the next valence band now crossing the EF to give a squarish hole pocket (red) centered at the Z point as seen in Fig.3(f).The imperfect FSN along the -Z direction now totally disappears, resulting in no maximum of Re() near the  point for BaGa4 in Fig.2(g).Although the Re &'( for BaGa4 is now at the Z point, there are no imaginary phonon modes as shown in Supplementary Figure2, thus no CDW either.These results from the calculated Re() and phonon dispersion show only CDW in SrAl4 but not the other three non-magnetic compounds with the same crystal structure and number of valence electrons, which agree well with experimental observations24 .
Fig.4(c) and (d) for comparison.As seen from the green shade for  NO , the largest difference is that SrAl4 (Fig.4(c)) has a much larger  NO than BaAl4 (Fig.4(d)) for the TA mode.Unlike SrAl4, the  NO of BaAl4 for the TA mode only has similar magnitude to the other modes near the  and Z points.The eigenvectors of the TA mode (Eu) and three such zone-center optical modes Eg (2.3 THz), B1g (6.9 THz) and A1g (10.8 THz) of SrAl4 are shown in Fig.4(e).The Eg mode corresponds to the in-plane motion of Al network, while the B1g and A1g modes at higher frequency correspond to out-of-plane motion of Al network.EPC interaction also directly affects the size of the phonon softening via the real part of phonon self-energy.The  NO decorated plots show that SrAl4 is subjected to a larger size of phonon softening than BaAl4.Given the small frequency of the TA mode near , the larger phonon softening in SrAl4 can induce this mode to be imaginary and give the CDW instability.As plotted in Fig.4(b) for the dispersion of TA mode with different electronic smearing, in contrast to BaAl4, this mode in SrAl4 has a substantial softening

4
(c)) to BaAl4 (Fig.4(d)), the highest phonon energy is reduced from 10.8 THz in SrAl4 to 9.9 THz in BaAl4 and the overall phonon energy range is rescaled by the mass factor as expected.Along the -Z direction, the longitudinal acoustic (LA) mode is also reduced from 3.0 THz in SrAl4 to 2.3 THz in BaAl4 at the Z point.However, the TA mode changes in the opposite direction and increases from1.4THz in SrAl4 to 1.8 THz in BaAl4 at the Z point as also shown in Fig.4(b).The first optical Eg mode at  for the in-plane motion of Al network also increases from 2.3 THz in SrAl4 to 2.7 THz in BaAl4.Secondly, comparing the TA mode lifetime  NO between BaAl4 and SrAl4 in Fig.4(a), the lifetime of the TA mode in BaAl4 becomes smaller than that in SrAl4 explained the origin of charge density wave (CDW) in SrAl4 and EuAl4 in comparison to the BaAl4, SrGa4 and BaGa4.Although all of them have nodal lines in the absence of spinorbit coupling (SOC) and become Dirac semimetals with SOC, the Dirac points are ~0.2 eV above the Fermi energy (EF).The Dirac-like dispersion of the valence and conduction bands form Fermi surface (FS) shells providing imperfect FS nesting (FSN) along the -Z direction for the three Al compounds, but not the Ga compounds due to the second valence band being pushed up to cross the EF.The susceptibility functions, (), have been calculated efficiently on a dense mesh of millions of k-points with the maximally localized Wannier functions (MLWF).The imperfect FSN is verified by the maximum of the real part of susceptibility function, Re &'(

Figure 1 .
Figure 1.Topological electronic band structures of SrAl4.(a) Crystal structure of SrAl4 in the body-centered tetragonal (tI10) of space group 139 (I4/mmm).Sr (Al) is in large green (small blue) spheres.(b) Bulk Brillouin zone (BZ) with high symmetry k-points and those on (001) surface BZ are labeled.A pair of Dirac points (DPs) along -Z direction are indicated by red dots.(c) SrAl4 band structure calculated in density functional theory (DFT) with spin-orbit coupling (SOC).The highest valence and lowest conduction bands are in blue and red, respectively.The dispersion along -Z is zoomed around the DP in (d) with the irreducible representations labeled and also the Al pz orbital projection at 4e site in green shade.(e) Nodal lines on the (110) plane shown by the zero gap.(f) Surface spectral function along the (001)  , - , direction.(g) (001) 2D Fermi surface at EF+0.21 eV with spin texture shown in green arrows.(h) Surface spectral function along the (001)  , - / direction showing the topological surface states converging to the projection of DPs in different directions together with (f).

Figure 2 .
Figure 2. Fermi surface, susceptibility functions, phonon dispersion and charge density wave of SrAl4.(a) SrAl4 Fermi surface (FS) consists of valence (yellow outside and blue inside) and conduction band (purple outside and green inside) from the side-view.The individual FS pieces of valence and conduction bands from the near top-view are in (b) and (c), respectively.(d) 3D real part of the susceptibility function (()) as in Re()/Re &'( of SrAl4 with the iso-surface value of 0.999 showing the maximum at q=±0.23(/c).(e) The imaginary part of the susceptibility functions Im()/Im &'( along the -Z direction for the four compounds.The vertical dashed line marks the experimental charge density wave (CDW) vector for SrAl4 at qCDW=0.22(/c).(f) 2D FS cut on the (110) plane showing the FS shells of the valence and conduction bands in SrAl4 with the nesting vector marked by an orange arrow.The high (low) intensity is in red (grey).(g) The real part of the susceptibility functions Re () /Re  &'( along the  -Z direction.(h) Phonon band dispersion () of SrAl4 with imaginary transverse acoustic (TA) mode along the -Z direction (black arrow).

Figure 3 .
Figure 3. Electronic band structure and Fermi surface of the other three alkali earth compounds.For the electronic band structure of (a) BaAl4, (b) SrGa4 and (c) BaGa4, the highest valence and lowest conduction bands are in blue and red, respectively.For the 3D Fermi surface (FS) of (d) BaAl4, (e) SrGa4 and (f) BaGa4, it consists of valence (yellow outside and blue inside) and conduction bands (purple outside and green inside) from the side-view.

Figure 4 .
Figure 4. Comparison of electron-phonon coupling between SrAl4 and BaAl4.Calculated phonon (a) lifetime  N and (b) dispersion  for the transverse acoustic mode along the -Z direction with different smearing in Ry for SrAl4 (solid lines) and BaAl4 (dashed lines).(c) and (d) Phonon band dispersion decorated with the mode-resolved electron-phonon coupling (EPC) strength ( N ) in green shadow for SrAl4 and BaAl4, respectively, with the electronic smearing of 0.04 Ry.(e) Eigenvectors of the four zone-centered phonon modes with sizable  N from low to high frequency as shown in (c) and (d).The atomic displacements are indicated by red arrows and the corresponding irreducible representations of Eu, Eg, B1g and A1g are labeled.

Figure 5 .
Figure 5. Band structure, electron-phonon coupling and bonding in EuAl4 with comparison to SrAl4 and BaAl4.(a) EuAl4 bulk band structure with the top valence and conduction bands plotted in blue and red, respectively.(b) Fermi surface (FS) consists of valence (yellow outside and blue inside) and conduction bands (purple outside and green inside) from the side-view.(c) 3D real part of susceptibility function Re /Re &'( with the maximum value shown as red pockets.(d) The 1D imaginary part of susceptibility function Im()/Re &'( plotted along the  -Z direction ( ) for SrAl4, BaAl4 and EuAl4.(e) Calculated phonon dispersion  of the transverse acoustic (TA) mode along the  -Z direction with an electronic smearing of 0.015 Ry for the three compounds.(f) The 1D real part of susceptibility function Re()/Re &'( .(g) The lifetime  N for the TA mode.(h-j) Electron charge density of ( \ ) − () − ( \ ) for X=Sr, Ba and Eu, respectively.The positive (negative) iso-surfaces of 2.1´10 -3 (e/Å 3 ) are shown in yellow (cyan) on the outside and blue from inside.