2 × 2 charge density wave in single-layer TiTe2

A density functional theory study concerning the origin of the recently reported charge density wave (CDW) instability in single-layer TiTe is reported. It is shown that, whereas calculations employing the semi-local functional PBE favor the undistorted structure, the hybrid functional HSE06 correctly predicts a distortion. The study suggests that the magnitude of the semi-metallic overlap between the valence band top at and the conduction band bottom at is a key factor controlling the tendency towards the distortion. It is also shown that tensile strain stabilizes a CDW, and we suggest that this fact could be further used to induce the instability in double-layers of TiTe, which in the absence of strain remain undistorted in the experiment. The driving force for the CDW instability seems to be the same phonon mediated mechanism acting for single-layer TiSe, although in single-layer TiTe the driving force is smaller, and the semimetallic character is kept below the transition temperature.


Introduction
Transition metal dichalcogenides of the groups IV and V rank among the most controversial materials exhibiting charge density wave (CDW) instabilities [1,2]. The possibilities of strong or weak electronphonon coupling scenarios in group V 2H-MX 2 ( M = Nb, Ta; X = S, Se) and either phonon mediated or excitonic mechanisms in group IV 1T-TiSe 2 have been discussed for decades [1]. Many of these systems also exhibit superconductivity (SC) under certain conditions and the competition between the two instabilities remains an important question still unanswered [3,4]. These materials are built from MX 2 layers interacting through weak van der Waals forces and thus are easily exfoliated [5]. Consequently, they offer the possibility to examine the above mentioned issues at the two-dimensional (2D) limit as well as by smoothly varying the density of carriers through gate doping. This is at the origin of the huge revival of interest recently raised by these materials [6][7][8][9][10][11].
Indeed, intriguing differences of these few-flake or even single-layer materials with their bulk counterparts have been discovered. Recent reports on the existence of a very weak pseudo-gap at the Fermi energy in single-layer NbSe 2 [6,12] or the possible occurrence of incommensurate modulations for slightly electron doped TiSe 2 crystals of thicknesses less than 10 nm [10,13] make clear that we are still far from a full understanding of the physics of CDW materials and more particularly when the screening is reduced.
In this context, the recent report of a 2 × 2 CDW in single-layer TiTe 2 by Chen et al [8] came as a very intriguing surprise. Since long [14][15][16] it has been known that bulk 1T-TiTe 2 does not exhibit the 2 × 2 CDW that occurs in isostructural 1T-TiSe 2 [17] In addition, the 2 × 2 CDW is not observed anymore in double-layer TiTe 2 [8]. In contrast, the 2 × 2 CDW instability is observed in TiSe 2 from the single-layer, for ultrathin films with up to six layers [18], and for the bulk crystal [17]. The experimental indication of the occurrence of the CDW in single-layer TiTe 2 is even more surprising when considering that first-principles density functional theory (DFT) calculations found that single-layer TiTe 2 shows no tendency to distort towards the 2 × 2 CDW structure at the generalized gradient approximation (GGA) level [8]. Yet, calculations of the same quality successfully predict that the 2 × 2 CDW structure is more stable than the nondistorted structure for single-layer TiSe 2 [13,18,19].

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Overall, these observations suggested the hypothesis that something really new and challenging is at work in single-layer TiTe 2 [8].
However, one should note that the CDW transition occurs at 100 K in single-layer TiTe 2 [8] but at a considerably higher temperature, 232 K, in single-layer TiSe 2 [11]. Hence, the driving force for the distortion must be considerably weaker in single-layer TiTe 2 . Before concluding that a new scenario is needed to grasp the origin of the unexpected 2 × 2 CDW in this mat erial, one should wonder about the appropriateness of the so far successful GGA-type of DFT approaches to the CDW instabilities in single-layer group IV and V dichalcogenides. Such an appraisal is needed because it impinges on very fundamental questions concerning CDW instabilities at the 2D limit. Note that it has been recently shown [20] that GGA-type functionals like PBE [21] overestimate the overlap between the Ti 3d and Se 4 p levels in bulk 1T-TiSe 2 . This can be corrected by using hybrid functionals like HSE06 [22,23] leading to an improvement of the electronic description of bulk TiSe 2 [20]. In the following, we report a DFT study of the likeliness of a 2 × 2 CDW in singlelayer TiTe 2 employing both PBE and HSE06 functionals which provides useful insight on the origin of the CDW instability in single-layer TiTe 2 .

Results and discussion
An isolated layer of 1T-TiTe 2 is made of an hexagonal lattice of Ti atoms in an octahedral environment of Te atoms ( figure 1(a)). The repeat unit of the hexagonal bulk crystal structure contains just one of these layers. A detailed description of our calculation method is presented in appendix. Let us start our analysis by briefly considering the PBE description of the electronic structure. The optimized a cell parameter is 3.804 Å for the single layer and 3.815 Å for the bulk, in good agreement with the bulk experimental value of 3.777 Å [24]. Within a single-layer there are several Te...Te contacts shorter than the sum of the van der Waals radii so that the valence bands, which have their maximum at Γ and are mostly built from Te 5 p orbitals, are considerably wide and overlap with the bottom part of the Ti 3d (~2%-3%) bands, which have their minima at the M point ( figure 1(b)). As expected from the fact that in the bulk there are short Te...Te contacts in the direction perpendicular to the layer, the semimetallic overlap is 19% larger in the bulk. Note that, as it is clear from figure 1(b), inclusion of spin-orbit coupling effects does not have any noticeable effect and therefore it will not be considered anymore in the following. Thus, according to the PBE calculations single-layer TiTe 2 is a semimetal exactly as the bulk. In contrast with the case of single-layer TiSe 2 , for which the same type of calculations led to a phonon with imaginary frequency at the M point [13], our calculations with the GGA functional for single-layer TiTe 2 (figure 1(c)) show no phonons with imaginary frequency, in agreement with those of Chen et al [8]. Although there is some remnant of the instability at M (notice the optical branch that disperses downwards and shows the lowest frequency at the M point, reminiscent of the mode that becomes unstable for TiSe 2 ), there is no definite indication of a phonon instability that may lead to the 2 × 2 distortion of the structure. Detailed structural optimizations of 2 × 2 supercells confirmed this result. However, as indicated by the frozen-phonon total energy calculation as a function of the soft phonon mode amplitude at M (figure 1(d)), the GGA potential energy surface is extremely flat. Under such circumstances, even if strictly speaking the PBE calculations disagree with the experimental results in that no tendency toward the 2 × 2 CDW distortion is found, the results are somewhat inconclusive and a closer look is needed.
The very flat frozen-phonon energy curve of figure 1(d) suggests that small external perturbations could be able to change the relative stability of the undistorted and 2 × 2 CDW structures. This could occur, for instance, by the effect of strain. Thus, we studied the evolution of the band structure and the relative stability of the undistorted 1 × 1 and 2 × 2 CDW structures as a function of biaxial tensile strain (allowing the atomic positions to relax for each strain applied). The strain is defined as s = δm/m 0 where m 0 is the unstrained cell parameter and m 0 + δm the strained cell parameter. Thus, positive values correspond to tensile strains. As shown in table 1, a tensile stress as small as 1% is sufficient to make the 2 × 2 CDW structure slightly more stable than the undistorted one. For strains larger than 2%, the 2 × 2 CDW is definitely favored.
The development of the 2 × 2 distortion is accompanied with the opening of energy gaps at the crossings between the folded valence and conduction bands. However, as shown in figures 2(a)-(c), the distorted structure is still semimetallic and only for relatively large tensile strains (between 5% and 6%) a full band gap occurs at the Fermi level. As shown in figure 2(a), in the folded band structure of the initial unstrained layer there is a band crossing slightly below the Fermi level at approximately halfway of the Γ-M line. Under a slight tensile strain, the semimetallic overlap decreases and this crossing occurs exactly at the Fermi level thus leading to the opening of a gap at that energy (see the folded and unfolded band structure for the stable 2 × 2 CDW structure under a 2% biaxial strain in figure S1 of the supplementary information (SI) (stacks.iop.org/TDM/6/015027/mmedia)). Nevertheless, the second Te-based valence band centered at Γ (the one with a larger effective mass) still crosses the Fermi level: it does hybridize with the M -point bands, but the resulting gap opens about 0.1 eV above the Fermi level. These results are in good agreement with the exper imental ARPES data [8], which indicate that the valence band with larger effective mass remains metallic for temperatures below the CDW transition T c , whereas the band with smaller effective mass does develop a gap at the Fermi level below T c . The calculated densities of states (DOS, figure S2 of the SI) reflect the same result, with a pseudo-gap occurring at the Fermi level for small strains under which the 2 × 2 CDW structure is already stable ( ∼ 2%-3%). This also compares favourably with the pseudo-gap observed in the experimental STS spectra (figure 4 in [8]), which has a similar width as the one observed in ARPES.
Judging from the comparison of the previous results with the ARPES data of Chen et al [8] it appears that the GGA-PBE description of single-layer TiTe 2 is only consistent with the experimental situation when a slight tensile strain is imposed in the calculation, as the GGA functional overestimates the overlap between conduction and valence bands, which is then compensated by the imposed tensile strain. The main effect of the strain is a decrease of the intralayer Te... Te short contacts which leads to a decrease of the Te 5 p bandwidth and, consequently, of the semimetallic overlap. Only when this overlap decreases with respect to the PBE description of the system the 2 × 2 CDW becomes more stable. This observation is consistent with the fact that, when Te...Te interlayer interactions come into play in the bulk or even in the double-layer, the semimetallic overlap increases and the 2 × 2 CDW is not observed anymore. As shown in figures 2(d)-(f) the semimetallic overlap increases from single-to double-layers. Note that the band structure for the TiTe 2 double-layer with ∼ 6% strain is very similar to that of the single-layer with a 3% strain.
These results are reminiscent of the above mentioned work concerning the functional type dependence of the semimetallic overlap of bulk TiSe 2 [20] and prompted us to reconsider the stability of the unistorted versus 2 × 2 CDW structures using the hybrid type functional HSE06 [22,23]. Shown in figure 3 is a frozen-phonon calculation of the energy difference between the undistorted structure and the 2 × 2 CDW structure following the soft phonon mode dist ortion. The curve clearly shows that, at the HSE06 level, unstrained single-layer TiTe 2 is indeed unstable towards the 2 × 2 CDW distortion. By relaxing the structure around the minimum of the frozen-phonon curve we have obtained an energy gain of 3.8 meV/ formula unit. This stabilization energy is much lower than the value obtained for a TiSe 2 monolayer using a PBE functional, 6 meV/f.u [13]. Hellgren et al [20] showed that for bulk TiSe 2 hybrid functionals predict a much higher stabilization energy than the PBE functional. Thus, it is understandable that according to our PBE type studies single-layer TiTe 2 does not tend to experience a 2 × 2 CDW instability. Overall, our data suggest a weak driving force for the distortion.
Since this is a significant result let us consider in more detail the region of the semimetallic overlap. A fatband analysis of the PBE and HSE06 band structures is shown in figures 4(a) and (b), respectively. A close inspection at the region around Γ points out some clear differences between the results of both functionals. For PBE, above the (partially empty) top of the valence band at Γ there is a non-degenerate band and a pair of degenerate bands at a higher energy. The order is the opposite for the HSE06 functional ( figure  4(b)), for which the two degenerate bands are lower in energy. Around the Fermi level a heavy mixing of the t 2g levels of the Ti atom and the 5 p levels of the Te atoms occurs. Assuming a local coordinate system in which the three-fold symmetry axis of the octahedron occurs along the z direction, the non-degenerate Ti-    the bottom of the conduction band at M for the PBE functional than for HSE06. The smaller overlap in the HSE06 band structure is also clear from the decrease of the the area of the constant energy plot at 0.25 eV below the Fermi level [8] for the undistorted TiTe 2 single-layer calculated with the HSE06 functional (compare figures S2 and S4 of the SI). We note that the PBE-type band structure for single-layer TiSe 2 , which provides a satisfactory description of the relative stability of the undistorted and 2 × 2 CDW in this system [13,19], shows exactly the same topology and particularly, the same band ordering as the HSE06 band structure of figure 4(b). As soon as one moves along the Γ-M line (i.e. along the a * direction), the only symmetry element preserved is the symmetry plane perpendicular to the layer and going along the a * direction. One of the two Ti-based doubly degenerate levels at Γ and the Ti d z 2 mix and interact with one of the Te p bands near Γ leading to the slowly descending band from Γ-M which is associated with the electron pockets near M . With the local system of axes mentioned above, the crystal orbitals around M are almost exclusively made of tilted Ti d x 2 −y 2 orbitals (i.e. a mixing of Ti d x 2 −y 2 and Ti d z 2 which leads to the tilting of the orbital) exactly as we recently reported for the TiSe 2 single-layers [13]. We refer the reader to this work for a detailed analysis of the nature of the band structure which entirely applies to the HSE06 one reported in figure 4(b).
The HSE06 band structure and density of states for both the undistorted and the 2 × 2 CDW structures are reported in figure 5. As shown in both figures the stabilizing 2 × 2 CDW distortion opens a gap at the Fermi level. The analysis of the nature and origin of the distortion goes along the same lines presented in detail in our recent work for the TiSe 2 single-layer and will not be repeated here [13]. However, we note that the non-activated conductivity is kept in single-layer TiTe 2 below the CDW transition temperature [8] so that no gap should occur at the Fermi level. Consequently, the HSE06 calculations exaggerate the tendency towards the transition. As mentioned above, the PBE calculations suggest that under reasonable tensile strain a 2 × 2 CDW is favored without the development of a band gap. Only when the strain is relatively large (i.e. around 5%-6%, see figure 2) a band gap really opens. The HSE06 band structure of figure 5(a) is similar to that in figure 2(c) corresponding to a 6% tensile strength. We believe that the HSE06 functional exaggerates the stability of the 2 × 2 CDW and that the real situation concerning the weaker stabilization of the CDW phase and the lack of a full bandgap opening would rather correspond to that of the PBE functional with a moderate tensile strain. This observation, together with previous results on bulk TiSe 2 [20] suggest that a predictive description of these TiX 2 systems is attainable by using the HSE06 functional and tuning the actual contribution of the exact exchange. However, from the viewpoint of the physical origin of the CDW we do not find any noticeable difference with the previously reported analysis of the 2 × 2 CDW instability of TiSe 2 single-layers.
To further assess our conclusion we carried out HSE06 calculations for a TiTe 2 double-layer. Two single-layers with the optimized structure were placed with the interlayer bulk distance. The stabilization energy of the 2 × 2 CDW is reduced to practically onehalf the value in the single-layer. Taking into account the small (and exaggerated) value for the single-layer, the driving force for the 2 × 2 CDW distortion in the TiTe 2 double-layer must be extremely small or most likely nil, as experimentally found.
Our study points out an interesting possibility. Since tensile strain has been found to be a useful technique to induce modifications in single-layer or fewflake materials [25][26][27][28], it is possible that the 2 × 2 CDW can be induced in double-layers or triple-layers of TiTe 2 by using a small tensile strain 3 . Another useful hint provided by our study is that the 2 × 2 CDW in single-layer TiTe 2 may be more stable and result with a band gap under tensile strain or may be suppressed under a slight compresive strain.

Conclusions
A density functional theory study concerning the origin of the 2 × 2 CDW distortion recently reported experimentally for single-layer TiTe 2 has been carried out. This report is surprising because neither doublelayer nor bulk TiTe 2 exhibit the 2 × 2 distortion and a PBE-based DFT study predicts that the undistorted structure is also more stable for the single-layer. Our study shows that, whereas calculations employing the semi-local functional PBE favor the undistorted structure, the hybrid functional HSE06 correctly predict the 2 × 2 distortion. However, the HSE06 calculations seem to exaggerate the stability of the distorted phase and, as a consequence, a noticeable band gap of more than 0.1 eV is induced at the Fermi level. This is in contrast with the metallic character of the TiTe 2 single-layers below the transition temperature. Interestingly, PBE type calculations for the case where the single-layer is subject to a slight tensile strain also favor the 2 × 2 distortion while keeping the semimetallic overlap. The study suggests that the magnitude of the semi-metallic overlap is a key factor controlling the tendency towards the distortion and consequently only functionals describing such overlap very accurately can provide a truly predictive description of the electronic structure.
According to the present study the mechanism of the CDW instability in single-layer TiTe 2 seems to be the same phonon mediated mechanism acting for single-layer TiSe 2 [13] although now the driving force is smaller and the semimetallic character is kept below the transition temperature. As mentioned above, the magnitude of the semimetallic overlap seems to be one of the key factors in controlling the likeliness of the 2 × 2 CDW. Taking into account that the overlap should increase in these TiX 2 systems when the number of short Te...Te contacts increase or when they become stronger, the overlap should increase from single-layers to bulk and from X = S to X = Te. Since the instability is not observed in bulk TiS 2 nor in doublelayer TiTe 2 , it seems that only a relatively narrow range of semimetallic overlaps is associated with the instability. In this respect, a significant result of the study is that tensile strain stabilizes the 2 × 2 CDW distortion in single-layer TiTe 2 . This could be used to induce the instability in double-or triple-layers of TiTe 2 which in the absence of strain remain undistorted, to induce a stronger distortion leading to the creation of a band gap in single-layer TiTe 2 or most likely to suppress the 2 × 2 CDW under a small compression. Such studies could provide useful insight on the CDW mechanism of group IV 1T-TiX 2 phases.