Evidence for nesting driven charge density waves instabilities in a quasi-two-dimensional material: LaAgSb2

Since their theoretical prediction by Peierls in the 30s, charge density waves (CDW) have been one of the most commonly encountered electronic phases in low dimensional metallic systems. The instability mechanism originally proposed combines Fermi surface nesting and electron-phonon coupling but is, strictly speaking, only valid in one dimension. In higher dimensions, its relevance is questionable as sharp maxima in the static electronic susceptibility \chi(q) are smeared out, and are, in many cases, unable to account for the periodicity of the observed charge modulations. Here, we investigate the quasi twodimensional LaAgSb2, which exhibits two CDW transitions, by a combination of diffuse xray scattering, inelastic x-ray scattering and ab initio calculations. We demonstrate that the CDW formation is driven by phonons softening. The corresponding Kohn anomalies are visualized in 3D through the momentum distribution of the x-ray diffuse scattering intensity. We show that they can be quantitatively accounted for by considering the electronic susceptibility calculated from a Dirac-like band, weighted by anisotropic electron-phonon coupling. This remarkable agreement sheds new light on the importance of Fermi surface nesting in CDW formation.

instability of a metallic phase against a spatial modulation of the free carrier density, and are generally accompanied with a static distortion of the crystal lattice. They have originally been predicted and reported in low-dimensional metals (e.g. blue bronzes [1], organic salts [2] or tritellurides [3,4]), but since then they have been reported in a vast variety of compounds encompassing 2D dichalcogenides [5], superconducting cuprates [6,7], or more recently in nickel pnictide superconductors [8]. The existence of CDW was originally proposed by Peierls who showed that a 1D chain is unstable due to a divergence of the static electronic susceptibility ( ) at the wave-vector = , that perfectly nests two parallel portions of the Fermi surface [9].
Through electron-phonon coupling (EPC), the phonon spectrum softens at , ultimately resulting in a static distortion of the lattice as a mode's energy vanishes. Unperfect nesting in real quasi-1D materials or in higher dimension compounds, on the other hand, rapidly supresses the susceptibility divergence [10], and thereby invalidates the Peierls scenario in the vast majority of CDW materials [11,12]. Even without resulting in a diverging electronic susceptibility, the presence of partial Fermi-surface nesting in dimension d>1 can enhance locally the EPC and eventually set the stage for the formation of a CDW [13]. Depending on the system, alternative approaches have been proposed to account for the formation of CDWs, which encompass strong momentum [14,15,16] or orbital [17] dependence of the EPC (in combination with strong anharmonic effects [18,19,20,21]) , spin fluctuations [22] or exciton condensation [23].
The family of RAgSb2, where R is a rare earth ion, has attracted growing attention in the last years due to the different low temperature ground states observed in these compounds when varying the R ion [24,25]. Bulk LaAgSb2 is a yet relatively unexplored compound of tetragonal structure (P4/nmm) that hosts two distinct CDW with two critical temperatures of = 207 K and = 186 K as evidenced by x-ray diffraction, transport, thermal and NMR studies [26,27]. The two CDWs present at low temperature in LaAgSb2 are aligned along the a and c axis, with a rather large real space periodicity of ~17 nm (CDW1, t1 ~ 0.026 a*) and ~6.5 nm (CDW2, t2 ~ 1/6 c*) and argued to be consistent with Fermi surface nesting [26]. Interestingly, recent magneto-transport [28] and ARPES [29]investigations have indicated that electronic bands in the vicinity of the Fermi energy involved in the formation of CDW1 are of Dirac type, dispersing linearly as a function of momentum as in graphene, albeit with a band velocity twice smaller. The estimated nesting vector ~(0.09±0.04) π/a, is large but relatively close to t1. Ab initio calculations further suggest that CDW2 is also related to nested parts of the Fermi surface associated with a distinct electronic band [25,26]. Time-resolved optical measurements revealed two low energy amplitude modes, suggesting that the CDW instability is triggered by the softening of a low-lying acoustic phonon mode [30]. To date, however, the dispersion of the phonons in this system and their possible role in the formation of the two CDW states has not been investigated.
Having in mind the aforementioned considerations regarding Fermi surface nesting driven formation of CDW in dimensions d > 1, we have performed a series of temperature dependent diffuse scattering (DS) and inelastic x-ray scattering (IXS) experiments in order to unveil the CDW formation mechanism in LaAgSb2. The wave vectors of the CDW instability can be directly identified in the normal state DS intensity distribution, and IXS investigations confirm the softphonon driven nature of the CDW instabilities. Interestingly, we observe that the complex 3D momentum distribution of the DS intensity accurately follows that of the 'partial' electronic susceptibility arising from the intraband contribution from the linearly dispersing electronic states.
These states do not form a complete Dirac cone (as the dispersion is linear only along one k-space direction, and parabolic in the orthogonal direction), but their strong nesting locally enhances the EPC and yields the CDW formation. This provides a textbook example for the CDW formation in higher-dimensional materials and suggests new routes for the possible design of such states through band structure engineering of metallic systems.
Bulk LaAgSb2 crystals were grown by the self-flux method [15]. DFT band structure calculations for the high-temperature tetragonal structure were performed with the mixed-basis pseudopotential method [35,36] in the local-density approximation [37]. Norm-conserving pseudopotentials were constructed following the description of Vanderbilt [38] and included the semi-core states La-5s, La-5p, Ag-4s, Ag-4p, and Sb-5s in the valence space. We used experimental lattice parameters taken from [39]  Interestingly here, we note that the tetragonal phase is stable against CDW instabilities in the calculation (that is all phonon frequencies remain real). As we shall see, the input from the calculation can be used to analyse the DS data and unveil the origin of the CDW formation in this system.
The DS study covered the temperature range from 100 K to room temperature (RT) with adaptive step from 5 to 20 K. The DS signal is seen in the entire reciprocal space, but can be best studied next to very weak Bragg reflections. In Fig On further cooling, a second set of CDW reflections -associated with CDW2 -appears at commensurate value t2 = c*/6. At the lowest investigated temperature (100 K) incommensurate satellites up to 6 th order are observable and combined satellites nt1+mt1'+pt2 are also clearly apparent (t1 and t1' are related by p/2 rotation around c*).  We have therefore unambiguously established that the strong temperature dependence of the DS above the CDW transition originates from the softening of phonons, which drives the CDW instabilities in LaAgSb2. Without any doubt, this strong anharmonicity is rooted in the EPC, but the origin of its momentum dependence remains to be clarified. In particular, given the quasi-two dimensionality of the system it is legitimate to wonder whether those can be associated with the Dirac-like band, as suggested in previous studies [26,29].
We argue below that a direct comparison between the momentum distribution of the DS intensity and the calculated electronic susceptibility of the normal state out-of-which the CDW emerge provides a straightforward way to settle this issue [40].
To do this, we have first calculated the real part of the static generalized electronic susceptibility: corresponds to a 'partial' susceptibility considering only the intra ( = ) and inter-( ≠ ) band contributions from each of the four bands crossing the Fermi level to the total susceptibility ( 9 ( ) is the Fermi-Dirac distribution, and the energy dispersion for each band is obtained from our DFT calculation). We recall here that the imaginary part of ( ), which involves only the electronic states at the Fermi level vanishes in the static limit due to causality [8]. Owing to the Kramers-Kronig relations, all electronic states from the dispersion contribute to the real part of ( ) which can therefore exhibit a richer momentum structure than that infered from nesting of the states at the Fermi level only.
As can be seen in Fig. 3b, the momentum dependence of ( ) is indeed quite complex, and in particular much richer than the DS patterns seen in Fig. 1. This is not surprising as DS does not directly probe ( ). All these )* ( ) have pronounced and specific momentum dependences but, remarkably, only '' ( ) shows a cross-like shape in the (h k 0) plane and a tubular structure along the c-axis, reminiscent of the DS patterns of Fig. 1. In Fig. 4, we compare in greater details the DS patterns around Γ '(( with the k-space structure of '' ( ). The precise location of the calculated intensity maxima slightly differs from those that are observed experimentally. This reminds of the mismatch between the proposed nesting vectors and t1 from previous studies [26,29], but this small difference can be accounted for by a minor (~70 meV) adjustment of the Fermi level, which is reasonable compared to the absolute precision of DFT calculations (initial EF = 0.188 Ry, became 0.183 Ry). Note that the shifted Fermi surface remains in very good agreement ARPES data from ref. [29] (see Appendix). Given the constrains imposed by these 'nesting surface shapes', which are way stronger than those derived just from the modulation vector, the excellent agreement we obtain here largely validates the adequacy of DFT calculation for the description of the band structure of LaAgSb2.
Importantly, all the features from our experimental data can be directly associated with '' ( ) and, conversely, none of them can be found in the other )* ( ) contributions. Note finally the small mismatch of the diffuse tube diameter in two orthogonal directions (a* and b* - Fig. 4), which is not seen in '' ( ) is naturally explained by the fact that the DS pattern is decorated by scattering selection rules from phonons [41], from which the fourfold symmetry is necessarily lost. This observation immediately raises the question of why should '' ( ) contribute to DS more than any other )* ( )?
We show that this naturally arises from the momentum dependence of the EPC, which we evaluate through that of the total (that is, summed over all branches) phonon linewidth )* ( ) = Taken together, these facts allow us to definitively claim that the formation of the CDW in LaAgSb2 is directly related to a nesting mechanism in which anisotropic EPC is rooted. It is important to highlight that this conclusion could not have been drawn from the analysis of the total electronic susceptibility ( ), in which the specific contribution of '' ( ) is averaged out. In other words, in a multiband system such as LaAgSb2, it is absolutely crucial to examine each )* ( ) contribution individually, as well as the momentum dependence of the corresponding EPC )* ( ) -rather than the total ( ) -to assess the relevance of nesting for the formation of the CDW.
We finally note that a comparison between Figs 1 and 4 also reveals that the tubular shape of the DS pattern along the c-axis is not affected by the soft-phonon condensation at TCDW1. This tubular structure is only visible in '' ( ), which is therefore also responsible for the formation of the CDW2 (but a different phonon branch is likely involved -its identification is however beyond the scope of the present study).
We end our discussion by noting that our result might appear in contradiction with the common wisdom that nesting is not relevant in d>1 materials [8,9]. Detailed analysis of the dispersion of the electronic states shown in Fig. 3g, which originate from the 5px and 5py states of

APPENDIX B: Calculated Fermi surface and comparison with ARPES data
In Fig. 7, we provide a direct comparison of our calculated Fermi surface with the experimentally determined one from ref. [29]. We show two cuts of the Fermi surface in the kz = 0 and kz = 0.5 planes. Our calculation for the later is strikingly similar that reported in [29], suggesting that the authors of this publication used a finite kz for comparison with the experiment (for which kz is always hard to define). The agreement for the Fermi surface sheets close to the X point appears better for the kz = 0 calculation, whereas the central pocket shape in the experiment seems closer to that calculated for kz = 0.5. In any event, the agreement between the calculation and the experiment is very good, and is not affected by the small 70 meV Fermi level shift we applied to reproduce our diffuse scattering data.

Figure 7
Comparison between our calculated Fermi surface (black lines) with that experimentally measured in [29]. We show two cuts of the Fermi surface in the kz= 0 and kz=0.5 planes for the original calculation (top line), and that for EF shifted by 70meV (bottom line), from which we obtain the best agreement with our diffuse scattering data.

APPENDIX C: Dirac-like point in LaAgSb2
In Fig. 8, we show the electronic band structure of LaAgSb2 in the vicinity of the Dirac-like points.
Band crossings with linear dispersive bands appear along the high-symmetry lines M-G and A-Z at points K1=(k1,k1,0) with k1=0.1920 and K2=(k2,k2,0.5) with k2=0.2028, respectively (see panels  Phonon frequencies and eigenvectors were obtained via density functional perturbation theory as implemented in the mixed-basis scheme [42]. In addition, this approach provides direct access to the scattering potential induced by a phonon, which is then subsequently used to calculate EPC