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 superconductivity1,2 or CDW with the latter to realize axion electrodynamics3 for the case of opposite chirality or a topological phase with monopole harmonic order4 for the same chirality. The Fermi surface nesting (FSN) among Weyl points has been recently studied for the CDW in the quasi-1D (TaSe4)2I5,6 and the long-wavelength helical magnetic order in NdAlSi7. Very recently CDW has been observed in the chiral CoSi8,9 with Kramer-Weyl points. CDW has also been used10 to engineer topological phases to realize Dirac points close to the Fermi energy (EF). However, besides FSN, the emergence of CDW also needs the important ingredient of electron-phonon coupling11,12 (EPC). The traditional view of CDW comes from Peierls’ instability13 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 materials14,15. Density functional theory16,17 (DFT) calculations18,19 and experiments20,21 on NbSe2 and TaSe2 have shown that the CDW in these compounds is determined by the real part of linear response or susceptibility function, Reχ(q), with a large EPC interaction to soften the acoustic phonon mode, while not by the FSN from the imaginary part of susceptibility function, Imχ(q), 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 structure22 have shown a range of interesting behaviors with a martensitic tetragonal-to-monoclinic structural transition in CaAl423, CDW in SrAl4 and EuAl424,25,26,27,28, and multiple magnetic transitions, topological Hall effects and Skyrmions in EuAl4 and related magnetic compounds at different temperature25,29,30,31,32,33,34,35. The CDW in SrAl4 and EuAl4 has been measured to have a small q-vector along the Γ-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 experiment24 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 mechanism14,15 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 Reχ(q) and Imχ(q) 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 χ(q), the FS shells give the maximum of the real part of the susceptibility function, Reχmax, along the Γ-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 theory36 (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 Reχmax along the Γ-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 Γ-Z direction, besides the Reχmax 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 experiment24. 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.

Results and discussion

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. 1a. The bulk first Brillouin zone (BZ) and (001) surface BZ with the high symmetry k-points are displayed in Fig. 1b. The two Al sites are non-equivalent with 4d at (0.5, 0.0, 0.25) and 4e at (0.0, 0.0, z). These Al sites form a network of buckled square lattices sandwiched between the electron-donating Sr layers at the 1a site. Such structure is the binary version of the Fe-based superconductors and the related 122 compounds, where the two non-equivalent 4d and 4e sites are replaced by 3d transition metals (Fe, Co, and Ni) and main group V elements (As, P), respectively. Noticeably, the shortest Al-Al distance of 2.598 Å (DFT 2.616 Å) is between the two 4e sites from the two neighboring buckled square lattices along the 4-fold c-axis, which is slightly shorter than the Al-Al distance of 2.690 Å (DFT 2.688 Å) among the same buckled square lattice (see Supplementary Table 1).

Fig. 1: Topological electronic band structures of SrAl4.
figure 1

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 Γ-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 Γ-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 are shown by the zero gap. f Surface spectral function along the (001) \(\bar{\varGamma }\)-\(\bar{X}\) direction. g (001) 2D surface Fermi surface at EF + 0.21 eV with spin texture shown in green arrows. h Surface spectral function along the (001) \(\bar{\varGamma }\)-\(\bar{M}\) direction showing the topological surface states converging to the projection of DPs in different directions together with f.

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The DFT-calculated band structure of SrAl4 is plotted in Fig. 1c. 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. 1e 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 Γ-Z direction toward the X point. More nodal lines and loops are plotted on (001) and (100) planes in Supplementary Fig. 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-orbitals37. With SOC, these crossings in nodal lines are all gapped out except for the one along the Γ-Z direction as zoomed in Fig. 1d. The point group symmetry is D4h. We have used Vasp2trace38,39 to analyze the elementary band representations. From the double group representation of D4h for the spinful system, the top valence band along the Γ-Z direction with the 4-fold rotation switches between two different 2-dimensioinal irreducible representations of Λ7 and Λ6, thus there is no mixing guarantied at the band crossing, or equivalently speaking, the pair of DPs is protected by the 4-fold rotational symmetry40,41,42,43,44,45. Our results also agree with the previous study22 on BaAl4 for the DPs along the Γ-Z direction.

The locations of the DPs are also indicated in the BZ by the red dots in Fig. 1b 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. 1f along the \(\bar{\varGamma }\)-\(\bar{X}\) direction. The spin texture in Fig. 1g shows that these surface states are spin-momentum locked as expected for topological non-trivial surface states. Furthermore, we have also plotted these topological surface states in the other direction of \(\bar{\varGamma }\)-\(\bar{M}\) in Fig. 1h 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.

Susceptibility functions and phonon dispersions

The 3D FS of SrAl4 is plotted in Fig. 2a. 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 Γ 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. 2b, c, respectively. To study FSN and CDW from band structure, the bare susceptibility function14, χ(q), based on DFT single-particle Kohn-Sham bands needs to be calculated in real and imaginary parts,

$${{{{\mathrm{Re}}}}}\chi \left({{{{{\boldsymbol{q}}}}}}\right)=\mathop{\sum}\limits_{n,m,{{{{{\boldsymbol{k}}}}}}}\frac{f\left({\varepsilon }_{n,{{{{{\boldsymbol{k}}}}}}}\right)-f\left({\varepsilon }_{m,{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}\right)}{{\varepsilon }_{n,{{{{{\boldsymbol{k}}}}}}}-{\varepsilon }_{m,{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}}$$
(1)
$$\mathop{{{{{\mathrm{lim}}}}}}\limits_{\omega \to 0}{Im}\chi \left({{{{{\boldsymbol{q}}}}}},\omega \right)/\omega =\mathop{\sum}\limits_{n,m,{{{{{\boldsymbol{k}}}}}}}\delta \left({\varepsilon }_{n,{{{{{\boldsymbol{k}}}}}}}\right)\delta \left({\varepsilon }_{m,{{{{{\boldsymbol{k}}}}}}+{{{{{\boldsymbol{q}}}}}}}\right)$$
(2)

where \(f\left({\varepsilon }_{n,{{{{{\boldsymbol{k}}}}}}}\right)\) and \(\delta \left({\varepsilon }_{n,{{{{{\boldsymbol{k}}}}}}}\right)\) are the Fermi-Dirac distribution and Delta functions, respectively, of εn,k the n-th band energy eigenvalue at the k point with the EF set to zero. The calculation of χ(q) requires very dense double meshes (k and q). Using the maximally localized Wannier functions46,47 (MLWF), we are able to calculate the 3D susceptibility function χ(q) efficiently with millions of k-points in the BZ.

Fig. 2: Fermi surface, susceptibility functions, phonon dispersion and charge density wave of SrAl4.
figure 2

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, c, respectively. d 3D real part of the susceptibility function (χ(q)) as in Reχ(q)/Reχmax of SrAl4 with the iso-surface value of 0.999 showing the maximum at q=±0.23(π/c). e The imaginary part of the susceptibility functions Imχ(q)/Imχmax along the Γ-Z direction for the four compounds. The vertical dashed line marks the experimental charge density wave (CDW) vector for SrAl4 at qCDW = 0.22(π/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 Reχ(q)/Reχmax along the Γ-Z direction. h Phonon band dispersion ω(q) of SrAl4 with imaginary transverse acoustic (TA) mode along the Γ-Z direction (black arrow).

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The real part of the 3D susceptibility function Reχ(q)/Reχmax of SrAl4 is shown in Fig. 2d for the whole BZ with its maximum appearing as two small red spheres along the Γ-Z direction with a small finite q-vector. As plotted along the Γ-Z direction in 1D in Fig. 2g, the calculated q-vector for Reχmax is 0.23(π/c) for SrAl4, agreeing well with the experimental data of 0.22(π/c) (dashed line) for the observed incommensurate CDW at 243 K24. Such a CDW q-vector has usually been associated with FSN, i.e., the sharp peaks in the imaginary part of susceptibility function, Imχ(q), especially for 1D cases. But as plotted in Fig. 2e along the Γ-Z direction, the peak in the Imχ(q) of SrAl4 around 0.23(π/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. 2b and is also indicated by an orange arrow in the 2D FS contour on the (110) plane in Fig. 2f. The calculated phonon band dispersion of SrAl4 is plotted in Fig. 2h showing an imaginary TA phonon mode (black arrow) along the Γ-Z direction with a q-vector of 0.24(π/c), which matches that of calculated Reχmax. This match of the q-vector of the Reχmax and imaginary TA mode in SrAl4 shows that Reχmax and the imperfect FSN can be an indicator for CDW. However, we will later show that imperfect FSN and the Reχmax 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. 3a, 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 Imχ(q) of BaAl4 as plotted in Fig. 2e, also has a maximum along the Γ-Z direction, which has a larger q-vector and with a more extended plateau than that of SrAl4. In contrast, the Reχmax of BaAl4 in Fig. 2g 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 Imχ(q) and Reχ(q) is not unexpected, because the latter include contributions from the bands away from the EF. As plotted in Supplementary Fig. 2, the phonon band dispersion of BaAl4 does not have an imaginary phonon mode or CDW near the Reχmax q-vector.

Fig. 3: Electronic band structure and Fermi surface of the other three alkali earth compounds.
figure 3

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. 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.

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Next for SrGa4 in Fig. 3b, e, besides the extra band inversion at the S point, the main difference is the upshift of the valence bands along the Γ-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 Reχmax q-vector along the Γ-Z direction for SrGa4 is reduced toward q = 0 as plotted in Fig. 2g. SrGa4 does not have a CDW either as shown by the absence of imaginary phonon dispersion in Supplementary Fig. 2. Moving to BaGa4 in Fig. 3c, the top valence band along the Γ-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. 3f. The imperfect FSN along the Γ-Z direction now totally disappears, resulting in no maximum of Reχ(q) near the Γ point for BaGa4 in Fig. 2g. Although the Reχmax for BaGa4 is now at the Z point, there are no imaginary phonon modes as shown in Supplementary Fig. 2, thus no CDW either. These results from the calculated Reχ(q) 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.

Electron-phonon coupling and CDW

Despite the similar band structure, FS and also the finite small q-vector for the Reχmax in BaAl4 when compared to SrAl4, the absence of CDW in BaAl4 demands a closer look. To reveal the origin of the observed CDW in SrAl4 but not BaAl4, using density functional perturbative theory (DFPT), we have calculated the phonon lifetime γqv due to EPC, i.e. the imaginary part of the phonon self-energy, which is proportional to both the EPC matrix elements \({\left|g\left(q\right)\right|}^{2}\) and Imχ(q). As plotted in Fig. 4a with different electronic smearing, the γqv for the TA mode along the Γ-Z direction in SrAl4 is two to three times larger than that in BaAl4. To put the EPC of the TA mode into perspective among all the other modes, γqv can be converted into the dimensionless mode-resolved EPC strength as in \({\lambda }_{{qv}}={\gamma }_{{qv}}/\left(\pi N\left({E}_{F}\right){\omega }_{{qv}}^{2}\right)\), where N(EF) is the electronic density of states at EF. With the electronic smearing of 0.04 Ry to stabilize the imaginary TA mode in SrAl4, the calculated λqv for both SrAl4 and BaAl4 are plotted on top of their phonon dispersion in Fig. 4c, d for comparison. As seen from the green shade for λqv, the largest difference is that SrAl4 (Fig. 4c) has a much larger λqv than BaAl4 (Fig. 4d) for the TA mode. Unlike SrAl4, the λqv of BaAl4 for the TA mode only has similar magnitude to the other modes near the Γ 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. 4e. 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 λqv 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 Γ, the larger phonon softening in SrAl4 can induce this mode to be imaginary and give the CDW instability. As plotted in Fig. 4b for the dispersion of TA mode with different electronic smearing, in contrast to BaAl4, this mode in SrAl4 has a substantial softening 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(π/c), in a good agreement with the experimentally observed qCDW24.

Fig. 4: Comparison of electron-phonon coupling between SrAl4 and BaAl4.
figure 4

Calculated phonon a lifetime γq and b dispersion ω for the transverse acoustic mode along the Γ-Z direction with different smearing in Ry for SrAl4 (solid lines) and BaAl4 (dashed lines). c, d Phonon band dispersion decorated with the mode-resolved electron-phonon coupling (EPC) strength (λq) 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 λq from low to high frequency as shown in c, d. The atomic displacements are indicated by red arrows and the corresponding irreducible representations of Eu, Eg, B1g and A1g are labeled.

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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 Γ-Z direction here. Firstly, comparing the phonon dispersion of SrAl4 (Fig. 4c) to BaAl4 (Fig. 4d), 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 Γ-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 from 1.4 THz in SrAl4 to 1.8 THz in BaAl4 at the Z point as also shown in Fig. 4b. The first optical Eg mode at Γ 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 γqv between BaAl4 and SrAl4 in Fig. 4a, the lifetime of the TA mode in BaAl4 becomes smaller than that in SrAl4 for both large and small q-vectors. Note the lifetime γqv accounting for the EPC interaction has no mass dependence through ωqν 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 Imχ(q). 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 Reχ(q). In Fig. 4a, lifetime γqv shows that the EPC interaction is two to three times smaller in BaAl4 than SrAl4 for the TA mode along the Γ-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 non-magnetic EuAl4 (no Eu 4 f ) in Fig. 5a, b resemble those of SrAl4 in Figs. 1c, 2a, 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 χ(q) as plotted in Fig. 5d, e with a slightly smaller nesting q-vector at 0.19(π/c) than SrAl4. Noticeably, the Reχmax of EuAl4 has a relatively narrow peak similar to SrAl4 at the small q-vector, rather than the extended plateau of BaAl4. The 3D Reχ(q)/Reχmax of EuAl4 plotted in the full first BZ in Fig. 5c shows the maximum is indeed only along the Γ-Z direction. For the TA mode dispersion plotted in Fig. 5e 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 Reχmax q-vector along the Γ-Z direction in Fig. 5f, the three EPC interactions of the TA mode are quite different. As plotted in Fig. 5g, the γqv of EuAl4 has an interesting behavior as a function of q when compared to SrAl4 and BaAl4. At small-q (<0.2), the γqv 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 γqv 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 Reχmax q-vector. Thus, the EPC strength shown by γqv 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 Reχmax q-vector.

Fig. 5: Band structure, electron-phonon coupling and bonding in EuAl4 with comparison to SrAl4 and BaAl4.
figure 5

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 Reχ/Reχmax with the maximum value shown as red pockets. d The 1D imaginary part of susceptibility function Imχ(q)/Reχmax plotted along the Γ-Z direction (q) for SrAl4, BaAl4 and EuAl4. e Calculated phonon dispersion ω of the transverse acoustic (TA) mode along the Γ-Z direction with an electronic smearing of 0.015 Ry for the three compounds. f The 1D real part of susceptibility function Reχ(q)/Reχmax. g The lifetime γq for the TA mode. (h–j) Electron charge density of \(\rho \left({{XAl}}_{4}\right)-\rho \left(X\right)-\rho \left({{Al}}_{4}\right)\) for X = Sr, Ba and Eu, respectively. The positive (negative) iso-surfaces of 2.1 × 10−3 (e/Å3) are shown in yellow (cyan) on the outside and blue from inside.

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The TA mode CDW here involves a local shear distortion perpendicular to the c-axis. 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 36.8 GPa 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 c-axis 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.

Table 1 Calculated elastic properties of SrAl4, BaAl4 and EuAl4 in density functional theory (DFT)
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To better understand these differences from electronic structure, we have plotted and compared the electron charge density difference of \(\rho \left({{XAl}}_{4}\right)-\rho \left(X\right)-\rho \left({{Al}}_{4}\right)\) for X = Sr, Ba and Eu, respectively in Fig. 5h–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, because the EPC matrix element calculation involves all the bands like the elastic property calculation, in contrast to the susceptibility function calculation involving only the bands near EF.

Conclusions

In conclusion, using the approaches based on density functional theory, we have 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 spin-orbit 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 Γ-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, χ(q), 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, Reχmax, along the Γ-Z direction with a small q-vector for the Al compounds, while it is absent in the Ga ones. Then among the three Al compounds, the electron-phonon coupling (EPC) calculations show a large EPC interaction at small q-vector for the transverse acoustic (TA) mode along the Γ-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 Γ-Z direction at small q-vector, besides the Reχmax 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.

Methods

Density functional theory16,17 (DFT) calculations have been performed with PBE48 exchange-correlation functional including SOC using a plane-wave basis set and projector augmented wave method49, as implemented in the Vienna Ab-initio Simulation Package50,51 (VASP). We use a kinetic energy cutoff of 300 eV, Γ-centered Monkhorst-Pack52 (11 × 11 × 11) k-mesh, and a Gaussian smearing of 0.05 eV. The ionic positions and unit cell vectors are fully relaxed with the remaining absolute force on each atom being less than 1 × 10−4 eV/Å. The DFT-calculated lattice parameters with PBE + SOC are within ~2% of the experimental data, showing good agreement (see Supplementary Table 1). Phonon band dispersions have been calculated with finite difference method using PHONOPY53 on a (3 × 3 × 4) supercell of the conventional cell with 360 atoms on the k-mesh of (4 × 4 × 2). To calculate susceptibility functions efficiently, maximally localized Wannier functions (MLWF)46,47 and the tight-binding model have been constructed to reproduce closely the band structure within ±1 eV of the Fermi level by using Group II sd and Group III sp orbitals. The susceptibility functions in Eqs. (1, 2) have been calculated with four bands, two valence and two conduction bands, around the EF on the dense (120 × 120 × 90) k and q-mesh using the MLWFs, where the Fermi-Dirac distribution is sampled at the temperature of 100 K and the Delta functions are approximated with Gaussian functions with a smearing of 0.02 eV. The surface spectral functions have been calculated with the surface Green’s function methods54,55 as implemented in WannierTools56. Electron-phonon coupling (EPC) interaction has been calculated with density functional perturbative theory (DFPT) as implemented in Quantum Espresso57 (QE) using ultra-soft pseudopotentials with a kinetic energy cutoff of 50 Ry, a (3 × 3 × 3) q-mesh and (9 × 9 × 9) k-mesh with different smearing. In Supplementary Fig. 3 we show the convergence of calculated phonon band dispersions of SrAl4 with respect to increased k-mesh to (18 × 18 × 18) and q-mesh to (4 × 4 × 4) in QE using DFPT and also a larger supercell of (4 × 4 × 4) of the conventional cell with 640 atoms and increased k-mesh in VASP using the finite displacement method with PHONOPY. The second-order elastic properties have been calculated using the stress-strain methodology58,59. The crystal structures and electron charge density difference have been visualized with VESTA60.