Abstract
Reducing the thickness of a material to its twodimensional (2D) limit can have dramatic consequences for its collective electronic states, including magnetism, superconductivity, and charge and spin ordering. An extreme case is TiTe_{2}, where a charge density wave (CDW) emerges in the singlelayer, which is absent for the bulk compound, and whose origin is still poorly understood. Here, we investigate the electronic band structure evolution across this CDW transition using temperaturedependent angleresolved photoemission spectroscopy. Our study reveals an orbitalselective band hybridisation between the backfolded conduction and valence bands occurring at the CDW phase transition, which in turn leads to a significant electronic energy gain, underpinning the CDW transition. For the bulk compound, we show how this energy gain is almost completely suppressed due to the threedimensionality of the electronic band structure, including via a k_{z}dependent band inversion which switches the orbital character of the valence states. Our study thus sheds new light on how control of the electronic dimensionality can be used to trigger the emergence of new collective states in 2D materials.
Introduction
Transition metal dichalcogenides (TMDs) offer a versatile platform to study the interplay of different collective quantum states^{1,2,3}. Excitingly, the charge density wave (CDW) phases in these materials have been shown to exhibit a strong dependence on material thickness down to the monolayer (ML) limit, with the emergence of modified ordering wavevectors^{4} and transition temperatures^{5}, and evidence for significant phase competition^{6}. A particularly intriguing case is that of TiTe_{2}. This is the sister compound of the famous charge density wave material TiSe_{2}^{7,8}, in which an unconventional (2 × 2 × 2) CDW emerges from a narrowgap semiconducting normal state^{9}. CDW order persists down to the monolayer limit, with a (2 × 2) ordering^{10}. In TiTe_{2}, a larger porbital valence bandwidth (due to the more extended Te 5p orbitals) and enhanced dp charge transfer (promoted by the lower electronegativity of Te) cause the Ti dderived conduction band and Te pderived valence band to substantially overlap (Fig. 1a–c). This semimetallic state persists in the bulk to low temperatures, and no CDW transition occurs^{11,12,13}.
Surprisingly, however, signatures of a CDW phase have recently been observed in monolayer TiTe_{2}, although not in bilayer or thicker films^{14}. This new CDW is associated with a (2 × 2) periodic lattice distortion, in striking similarity to the CDW observed in MLTiSe_{2} (Fig. 1e–g), despite the significant band overlap and semimetallicity in the normal state of the Tebased system. The origins of this instability, and why it is only stabilised in the monolayer limit, have remained elusive to date. Here, we study the CDW transition in MLTiTe_{2} using temperaturedependent angleresolved photoemission spectroscopy (ARPES), hybrid density functional theory (DFT) calculations, and minimalmodel based approaches. Through this, we find evidence of a marked but symmetryselective band hybridisation between the valence and conduction bands which ultimately provides the energetic incentive to stabilise the CDW phase. For the bulk system, we demonstrate how pronounced outofplane band dispersions lead to mismatched Fermi surfaces and inverted valence band symmetries. As a result, the corresponding electronic energy gain is highly suppressed, explaining the marked difference in the ground state properties of single vs. multilayer TiTe_{2}.
Results
CDW state in MLTiTe_{2}
To confirm the previously reported CDW state in MLTiTe_{2} as an intrinsic instability of the perfect ML, we start with DFT calculations of the lattice dynamics. The calculated phonon dispersions of bulk 1TTiTe_{2} are shown in Fig. 1d. These show an absence of any imaginary frequency modes: the 1T crystal structure thus is predicted to remain stable down to 0 K. This is consistent with the absence of any CDW transition in our measurements of the electronic structure from bulk TiTe_{2} in Fig. 1b and with the previous experiments^{11,12,13}. In contrast, for the monolayer (Fig. 1h), our calculations indicate that the 1T crystal structure is no longer stable. The imaginary frequency mode at the M point indicates the tendency of the structure to undergo a (2 × 2) lattice distortion. This phonon mode has an A_{u} irreducible representation at M, consistent with the soft phonon mode thought to underpin the tripleq CDW instability in MLTiSe_{2}^{15}. Tracing the stability of this mode (Fig. 1i), we find a classic anharmonic doublewell potential, confirming the intrinsic instability of MLTiTe_{2} to undergoing a periodic lattice distortion upon cooling.
The underlying lattice instability is qualitatively the same as that in the sister compound MLTiSe_{2} (Fig. 1i), but we find that the depth of the potential well in the telluride is much smaller. This indicates that the instability in MLTiTe_{2} is intrinsically weaker than that of TiSe_{2}, where signatures of a strongcoupling CDW transition have been observed^{10,16}. Nonetheless, consistent with a recent work^{14}, we find that the new periodic potential arising from the softening of the Mpoint phonon mode in MLTiTe_{2} is sufficiently strong to induce a ‘backfolded’ copy of the valence band dispersions, evident at the original M point in our measurements of the lowtemperature electronic structure (Fig. 1f). The backfolded spectral weight is weaker than the corresponding spectral signatures in MLTiSe_{2}^{16}, qualitatively consistent with a ‘weaker’ instability in the telluride as discussed above. Nonetheless, its observation points to a clear electronic component to the predicted periodic lattice distortion.
To investigate this in detail, we study the temperaturedependent evolution of the electronic structure of MLTiTe_{2} in Fig. 2. With decreasing temperature, the spectral weight of the characteristic backfolded valence bands at the Mpoint gradually increases (Fig. 2a). This is evident in the spectral weight extracted from fits to momentum distribution curves (MDCs) at E − E_{F} = −0.14 eV (Fig. 2b, see Supplementary Fig. 1 for the fitting details), where a rapid increase of the backfolded weight is evident at a temperature T ≈ 110 K. This is similar, although slightly higher, than the value of 92 K reported by Chen et al.^{14}, and we assign this as the CDW transition temperature in this system, T_{CDW}. We note that there is residual spectral weight evident at temperatures above this transition in Fig. 2b. This likely reflects finite background intensity in the fits from the tail of the conduction band states, rather than signifying precursor backfolding above the ordering temperature, as was found, for example, in the selenide case^{16}.
The CDW critical temperature we observe is substantially lower than T_{CDW} ≈ 220 K in MLTiSe_{2}^{16}. This is qualitatively consistent with the calculated doublewell potentials shown in Fig. 1i: the depths of these wells are related to the thermal energy at which the distorted phase is no longer stable as compared to the hightemperature structure, located at the saddle point of these wells. The smaller depth of the potential well in the case of MLTiTe_{2} than TiSe_{2}, therefore, indicates that the CDW ordering would be expected to onset at a lower temperature in the former, as observed here.
The (2 × 2) order predicted by our calculations is in agreement with a superstructure modulation imaged by scanning tunnelling microscopy in ref. ^{14}. This naturally explains the valence band backfolding from the Γ to the Mpoint of the Brillouin zone, although a concomitant backfolding of the conduction band from M to Γ has not been observed to date. Here, we find the appearance of weak spectral weight in between the innermost valence band in the ordered state (Fig. 2c). Its spectral weight grows with decreasing temperature, as evident from a peak at ≈110 meV binding energy in energy distribution curves (EDCs) taken at the Brillouin zone centre (Fig. 2d (see also Supplementary Fig. 2 for peak fitting analysis and for further discussion on the lineshape and temperature dependence of this component). A secondorder derivative analysis (inset of Fig. 2c) indicates how this additional spectral weight derives from an electronlike band, suggesting its origin as due to a backfolding of the conduction bands in the CDW phase of MLTiTe_{2}.
Orbitalselective band hybridisation
The secondderivative analysis also suggests the opening of spectral gaps in the reconstructed electronic structure. To investigate this in detail, to understand better how the electronic structure around the Fermi level evolves across the CDW transition, we performed highresolution measurements at the Γ point of the Brillouin zone. Here, the spectral weight is dominated by the valence bands (Fig. 3a). Comparing the high and lowtemperature spectra, a small spectral gap can be seen to open in the vicinity of the Fermi level for the outer valence band, while no such gap is obvious for the inner valence band. To probe this quantitatively, we show the amplitude (height) of Lorentzian fits to MDCs of the outer and inner valence bands in Fig. 3b, c, respectively (see also Supplementary Fig. 3 for details of the fitting). The intensities vary smoothly for the hightemperature measurements, with an overall decrease in intensity approaching the Fermi level reflecting the effects of transition matrix elements on the photoemission measurements, and also the influence of spectral broadening. This global intensity variation is still present at low temperature, but with an additional modulation leading to dips in fitted intensity at binding energies of ≈50 meV and ≈100 meV for the outer and inner bands, respectively. This is clearly visible in normalised difference plots of the fitted intensity at high and low temperatures (coloured arrows in Fig. 3d, e). The suppressed spectral weight occurs at the energies at which the original valence and conduction bands intersect when backfolded in the lowtemperature phase (inset of Fig. 3b). This suggests their origin is due to a band hybridisation between the backfolded conduction and valence bands in the CDW state. Spectral broadening has precluded the observation of such hybridisation signatures to date; here, it still leads to remnant intensity, with the spectral weight within the hybridisation gaps not reducing to the background level, but nonetheless, the characteristic dips in spectral intensity point to clear evidence for the opening of band gaps in the electronic structure. Intriguingly, however, our measurements indicate that these gaps are highly asymmetric, with a pronounced minimum evident for the outer band, while only a small dip is observed at the band crossing energies for the inner valence band.
We will show below that this reflects an orbital and symmetryselective band hybridisation, providing key insight into the stabilisation of CDW order in MLTiTe_{2}, and its absence in the bulk. To demonstrate this, we introduce a minimal 5band model capturing the lowenergy electronic structure of MLTiTe_{2} in the CDW state (see Methods):
where e_{i} {i = 1, 2, 3} represent the normal state dispersion of the elliptical electron pockets centred at neighbouring M points which are backfolded to Γ in the lowtemperature phase (Fig. 1(g)), h_{i} {i = out, in} are the two hole bands centred at Γ, spinorbit coupling (SOC), λ_{SO}, is included for the Tederived hole bands, and c(x) and s(x) represent \(\cos (x)\) and \(\sin (x)\), respectively.
The key physics of the resulting band hybridisation between the backfolded conduction and valence states is included via an interaction term Δ, whose strength is modulated by angledependent form factors, where θ is the angle of the momentum vector within the 2D plane, \(\theta ={\tan }^{1}({k}_{y}/{k}_{x})\). We note that Δ here does not distinguish between different microscopic mechanisms (e.g. electron–phonon coupling, electronhole interactions) which are ultimately required to drive the CDW and associated periodic lattice distortion. Rather it encodes the changes in the lowenergy electronic structure which result from, and can thus ultimately enable, such mechanisms to drive formation of a CDW state. The allowed band structure changes are further strongly constrained by symmetry, with the angledependent form factors in our model stemming from the particular orbital textures of the valence states. Along ΓM, our tightbinding calculations, informed from our DFTcalculated electronic structure and optimised to match the experimental normalstate dispersions (see Methods), indicate that the inner valence band is derived mainly from Te 5p_{x} orbitals, while the outer valence band has a dominant p_{y} orbital character, where x and y are referenced to the global coordinate system (Fig. 4d). Moving away from this direction, the orbital content rotates such that, for the perpendicular ΓK direction, the inner band is predominantly of p_{y} character, while the outer band is p_{x}like. The inner and outer valence bands thus host radial and tangential orbital textures as shown in the inset of Fig. 4a, b, much like those recently uncovered in, e.g., topological states of Bi_{2}Se_{3}^{17,18} and Rashba states of BiTeI^{19}. Additionally, including spinorbit coupling (SOC) for the Te 5p manifold in our tightbinding modelling leads only to a partial mixing of the orbital character between the inner and outer valence bands; the dominant tangential and radial orbital character remains largely unchanged at the Fermi level (see Supplementary Fig. 4 for tightbinding calculations including SOC). We show below that these orbital textures lead to a strong momentumdependence of the allowed band hybridisation in the ordered state of TiTe_{2}.
To demonstrate this, we consider the overlap of the p states with one of the three backfolded conduction band pockets, e_{1}, which has predominantly \({d}_{x^{\prime} y^{\prime} }\) symmetry in the octahedral basis ({\(x^{\prime} ,y^{\prime} ,z^{\prime}\)}, Fig. 4d)^{15}. The situation for the other bands follows from the threefold rotational symmetry of the system. To understand the hybridisation, we consider two core symmetries of the normalstate 1T crystal structure: a mirror plane, m, that is oriented along ΓM, and a C_{2} rotational symmetry axis oriented along ΓK. The 3\({d}_{x^{\prime} y^{\prime} }\) orbital has even parity in both of these (see inset in Fig. 4c), consistent with the A_{g} irreducible representation of the electron pocket at M^{20}. In contrast, the state made from p_{y} orbitals on the two chalcogen sites is even in C_{2}, but odd in m, while the opposite is true for the p_{x}derived bands (see insets in Fig. 4c).
Considering at the Γ point, where both of these symmetries are present for the hightemperature 1T crystal structure, the overlap integrals \(\left\langle {p}_{x} {d}_{x^{\prime} y^{\prime} }\right\rangle\) and \(\langle {p}_{y} {d}_{x^{\prime} y^{\prime} }\rangle\) are strictly zero, as a result of the opposite parity of these states under the C_{2} rotation and the mirror symmetry, respectively. At the CDW transition, however, the softening of three A_{u} phonon modes (Fig. 1h) leads to a periodic lattice distortion^{15} which breaks the mirror symmetry, while preserving the C_{2} rotational symmetry (the corresponding atomic displacements arising from the softening of one of the three phonon modes are indicated by the black arrows in the insets of Fig. 4c). The hybridisation of the p_{x} and dderived states, \(\left\langle {p}_{x} {d}_{x^{\prime} y^{\prime} }\right\rangle\), remains zero due to the C_{2} symmetry. However, the mirrorsymmetry enforced constraint of a lack of hybridisation between the p_{y} and dderived states in the normal state is relaxed in the distorted structure. This is evident from our calculated overlap integrals in Fig. 4c, where the hybridisation matrix element between p_{y} and \({d}_{x^{\prime} y^{\prime} }\) tesseral harmonics increases linearly with the amplitude of the atomic displacements within the periodic lattice distortion, while \(\left\langle {p}_{x} {d}_{x^{\prime} y^{\prime} }\right\rangle\) remains zero irrespective of the atomic displacement.
While these arguments are formally valid only at Γ where both the symmetries are present, they provide the basis for an understanding of the band hybridisation throughout the Brillouin zone, where the C_{2} (in both the normal and lowtemperature states) and m (in the normal state) symmetries are present along the entire ΓK and ΓM directions, respectively. In particular, they indicate that new channels for band hybridisation between the backfolded valence and conduction band pockets are only opened up by the periodic lattice distortion where conduction bands of \({d}_{x^{\prime} y^{\prime} }\) character intersect valence bands of p_{y}orbital character (and symmetryequivalent versions of these for the other conduction band pockets).
Combined with the momentumdependent orbital textures of the valence bands discussed above, this significantly constrains and modulates the hybridisation in the 2D kspace, as captured in the minimal model of Eq. (1), and shown from our model calculations in Fig. 5a, b, d. For example, these symmetry constraints prohibit the hybridisation of the e_{1} and h_{in} bands, while a gap would be expected to open at the crossing of the e_{1} and h_{out} bands. In reality, this strict symmetry protection is weakened by the inclusion of spinorbit coupling, which partially mixes the Te p_{x} and p_{y} states. As a result, even for the crossing of e_{1} and h_{in}, a small gap is opened (SOC is fully included in Fig. 5d; additional calculations in Supplementary Fig. 5 reveal the strict protection of the crossing in the absence of SOC). However, the gap structure remains strongly asymmetric, reflecting the symmetryselectivity discussed above, with a density of states which is extremely similar to the case without SOC (Supplementary Fig. 5). The overall hybridisation scheme derived from our model is qualitatively consistent with the results of recent DFT calculation of MLTiTe_{2} in the (2 × 2) phase^{21,22}: although the energy gap is overestimated in such calculations, they predict a weakly hybridised band laying in between the CDW gap as in our model.
To further validate our model and benchmark it against our experimental data, we simulate the resulting ARPES spectra expected from these hybridised bands (as described in Methods). We find a good agreement between our simulated and measured spectra (Fig. 5c, e) taking Δ = 42 ± 10 meV. This is a significantly weaker interaction strength than for the sister compound MLTiSe_{2}, where Δ ≈ 100 meV can be directly estimated from the experimental data^{16}, entirely consistent with the weaker nature of the instability predicted by our calculations of the lattice dynamics discussed above (Fig. 1i) and with the lower T_{c}. Crucially, our simulated spectra well reproduce the suppression of spectral weight in the ordered phase identified in fits to MDCs of our measured spectra (see green lines in Fig. 3d, e), including the pronounced asymmetry in spectral weight suppression at the crossing of e_{1} and h_{in} and h_{out}. This, therefore, provides direct experimental evidence for the symmetry and orbitalselectivity of the band hybridisation at the CDW phase transition in MLTiTe_{2}.
CDW energetics: from monolayer to bulk
Having validated the essential properties of our model, we can use this to provide new insight on the key question of why the CDW phase becomes stable in MLTiTe_{2}, while it is absent for the bulk. First, we note that there is in fact a significant electronic energy gain that results from the above band hybridisations at the CDW transition in the monolayer system. While the hybridisation gaps along ΓM occur largely below the Fermi level, at other angles around the Fermi surface, the gaps open at the Fermi level itself (Fig. 5a) partially gapping the Fermi surface. Indeed, calculating the density of states for the normal and hybridised states (Fig. 5f, shown over an extended energy range in Supplementary Fig. 6) it is evident how the hybridisation in MLTiTe_{2} significantly lowers the total electronic energy over an extended bandwidth comparable to the band overlap in the normal state. We thus conclude that this electronic energy gain ultimately enables the CDW transition to occur in MLTiTe_{2}.
Given this, one might assume that a similar band hybridisation in the bulk would enable the stabilisation of the CDW there, where none is known to exist experimentally, nor is one found as an instability in our DFT calculations (Fig. 1d). However, there are two key distinctions in bulk TiTe_{2} as compared to the ML case. First is a sizeable outofplane dispersion of the electronic states, for both the valence and conduction bands, as shown in Fig. 6a. The conduction band states yield large Fermi pockets for k_{z} = π/c, while they are barely occupied for k_{z} = 0. Though the valence state dispersion is weaker, the Fermi wavevectors nonetheless exhibit a notable decrease with increasing k_{z}. For a (2 × 2) ordering as in the monolayer case, the backfolded electron and holelike Fermi pockets are completely mismatched in size, and thus unlikely to lead to any significant electronic energy gain as for the monolayer. Even for a (2 × 2 × 2) instability as in the sister compound TiSe_{2}^{7}, the pockets are still significantly mismatched (Fig. 6a), causing the energies at which the hypothetical backfolded bands overlap to disperse strongly as a function of the outofplane momentum. Including a hybridisation between these bands assuming the same orbital texture as found in the monolayer case again leads to characteristic dips in the DOS, but these DOS suppressions now move from the occupied to the unoccupied states as a function of the outofplane momentum, k_{z} (Fig. 6b). As a result, this significantly reduces the effectiveness of the band hybridisation here to lower the total electronic energy of the system as compared to the normal state as shown in the total DOS in Fig. 6c.
Moreover, our DFT calculations indicate that the significant outofplane dispersion in fact leads to a k_{z}dependent band inversion of the d and pderived states along the ΓA direction, of the form known from other families of TMDs where it generates bulk Dirac points and topological surface states^{3}. As shown in Fig. 7a, this causes the orbital character of the inner and outer valence bands to become inverted between the Γ and the Aplane, switching approximately midway through the bulk Brillouin zone along k_{z}. In addition, p_{z} character is also mixed with the p_{x} states. The p_{z} orbitals exhibit odd parity with respect to the C_{2} axis, forbidding their hybridisation with the electron pockets as for the p_{x}derived states. The inplane rotational symmetry of the p_{z} orbitals is also reflected via an isotropic p_{z}dominated character of the outer Fermi surface sheet for k_{z} = 0, shown in Fig. 7b, c. This suppresses the hybridisation with the electron pockets even away from the ΓM direction.
Incorporating all of these effects (k_{z}induced size mismatch, switch of orbital selectivity mediated by the band inversion, and the additional incorporation of p_{z} character) into our minimal model (see Methods), we obtain a more realistic representation of the hypothetical symmetryallowed band hybridisation that would occur if the bulk system were to undergo a (2 × 2 × 2) periodic lattice distortion. In such a case, taking the same hybridisation strength as we found for MLTiTe_{2}, we find that only minimal changes in the occupied DOS occur as compared to the normalstate for bulk TiTe_{2} (Fig. 7d). Our calculations thus demonstrate that the electronic stabilisation of the CDW phase present in the monolayer is effectively completely removed in the bulk case.
Discussion
Our work thus points to a key role of orbitalselective band hybridisation in dictating whether CDW instabilities can occur in TiTe_{2}. We note that the electronic energy gain resulting from such band hybridisation is a necessary but not sufficient criterion for realising the CDW, and there still needs to be a microscopic interaction which can ultimately trigger the transition. In the sister compound TiSe_{2}, there has been significant discussion of the possibility that the transition could be driven by, or strongly enhanced by, electronhole interactions, with the resulting CDW discussed as a possible realisation of an excitonic insulator^{23,24,25,26}. The excitonic mechanism seems unlikely to play a significant role in the semimetallic telluride case considered here, due to the large overlap between the conduction and valence bands and the strong electronic screening that the free carriers provide. This is expected to significantly suppress the electronhole interaction, and is consistent with the persistent metallicity that we find in the CDW state: the symmetryallowed band hybridisations lead to momentumdependent gap formation at the Fermi level, leaving a welldefined Fermi surface even in the lowtemperature phase. Together, these factors suggest that the CDW transition studied here is driven by electron–phonon coupling. This is, however, strongly entangled with the arguments based on electronic energy gain here. Indeed, the symmetry of the soft phonon mode of the parent structure as determined by our DFT calculation directly constrains the allowed band hybridisation, showing how new hopping paths are enabled by the corresponding atomic displacements.
Our experimental measurements and model calculations indicate how significant electronic energy gains result from such a band hybridisation for the (2 × 2) lattice distortion in the monolayer case. However, we have found how these are almost completely suppressed due to the threedimensionality, and associated band inversions, in the bulk case. This provides a natural picture to explain the emergence of CDW order in monolayer TiTe_{2} as well as its intriguing thicknessdependence, and may bring new insight to understand a recently observed enhanced CDW order in moiré superlattices of TiTe_{2}based heterostructures^{27}. More generally, the model developed here will provide a natural framework through which to interpret and understand the lowenergy electronic structure evolution in the sister compounds TiSe_{2} and ZrTe_{2}, which host the same crystal structure and similar electronic structures, and where the nature of their CDWlike order is under intense current debate^{8,26,28}. In this respect, we note that the smaller coupling parameter determined here for MLTiTe_{2} than is applicable to MLTiSe_{2}^{16} points to a weaker microscopic driving force in the telluride system. Further tailoring the band structure of group IV dichalcogenide monolayers by synthesising mixed anion TiTe_{x}Se_{(2−x)} compounds would provide an ideal platform in which to study the transition between these two regimes, and to determine and disentangle how their propensity for CDW order is affected by variations in the low energy band structure and electronic screening.
Methods
Sample preparation
MLTiTe_{2} was grown by molecular beam epitaxy (MBE) on bilayer grapheneterminated SiC wafers using the method described in ref. ^{29}. The bilayer graphene termination of the SiC wafer was achieved by direct current heating of SiC wafers at 1500 ^{∘}C in a dedicated highvacuum chamber equipped with a pyrometer to check the temperature. These substrates were degassed at 600 ^{∘}C before the growth was commenced. The epitaxy of MLTiTe_{2} was performed in a highly Terich environment (Ti:Te flux ratio ~10^{3}) at a substrate temperature of 400 ^{∘}C as measured by a thermocouple placed behind the substrate. The growth was stopped after 70 min while the graphene RHEED streaks are still visible (Fig. 8a, b) and before the onset of bilayer formation, as confirmed from the lack of any bilayer splitting in our subsequent ARPES measurements of the electronic structure.
Supplementary Fig. 7 shows the corresponding lowenergy electron diffraction (LEED) pattern of the sample measured after growth. This shows sharp TiTe_{2} diffraction peaks in the radial direction, but with a partial rotational disorder, common in epitaxial TMD monolayers grown on weakly interacting substrates^{6}. The sample was then capped with Te for 20 min resulting in a spotty RHEED pattern (Fig. 8c), indicating a crystalline Te layer completely covering the underlying TiTe_{2} monolayer. An additional amorphous Se capping was deposited to protect the Te layer against oxidation in air, and was deposited until the Te signal disappeared completely (Fig. 8d). After growth, the film was capped with an initial Te layer and a subsequent amorphous Se layer which were removed by in situ annealing in the CASSIOPEE endstation at SOLEIL synchrotron immediately prior to performing ARPES measurements. Additional measurements on bulk TiTe_{2} single crystals were performed on freshly cleaved singlecrystal samples.
DFT
DFT calculations were performed using the Vienna Ab Initio Simulation Package (VASP)^{30,31,32} in order to investigate the lattice dynamical properties of TiSe_{2} and TiTe_{2} (bulk and monolayer). The interactions between the core and valence electrons were described using the projector augmented wave (PAW) method^{33}, with Ti 3p^{6}4s^{2}3d^{2}, Se 4s^{2}4p^{4} and Te 5s^{2}5p^{4} treated as valence electrons. The HSE06 (HeydScuzeriaErnzerhof)^{34,35} screened hybrid functional was employed to describe the exchange and correlation functional. Within HSE06, the exchange interaction is split into shortrange (SR) and longrange (LR) parts, with 25% of the SR exchange modelled using the exact nonlocal Fock exchange and the remainder of the contributions coming from the PBE (PerdewBurkeErnzerhof)^{36} functional. The hybrid functional calculations are computationally very demanding, but are expected to provide a better description of structure and phonon frequencies^{37}. The empirical correction scheme of Grimme’s DFTD3 method^{38} was also employed along with the hybrid functional to model the van der Waals interactions between the layers, as it has been successful in accurately describing the geometries of several layered materials in the past^{39,40,41}. For the monolayers, a vacuum layer of 12 Å was included to avoid interaction between the periodic images. Geometry optimisations were performed by setting the planewave energy cutoff at 350 eV for each structure, alongside Γ centred kpoint meshes of 16 × 16 × 10 (TiSe_{2}bulk), 16 × 16 × 8 (TiTe_{2}bulk), and 16 × 16 × 1 for the monolayers. A full relaxation of atomic positions, cell volume and shape was performed for the bulk structures until the residual forces acting on all the atoms were found to be <10^{−4} eV/Å. The same convergence criterion for forces was used for relaxing the internal atomic coordinates of the monolayers. The selfconsistent solution to the KohnSham equations was obtained when the energy changed by <10^{−8} eV. The phonon calculations were performed using the finitedisplacement method (FDM)^{42,43} as implemented in the Phonopy package^{44,45}. The kpoint sampling meshes were scaled accordingly as appropriate for the supercell calculations. We also mapped the potential energy curves spanned by the imaginary frequency modes in the undistorted structures to estimate the barrier associated with CDW transition. For this purpose, we used the ModeMap code to create a series of structures with atomic displacements along the mode eigenvectors over a range of amplitudes "frozen in". This was followed by performing singlepoint energy calculations on the modulated structures to obtain the doublewell potential spanned by each mode.
Tightbinding
The band structure of MLTiTe_{2} was calculated adapting the SlaterKoster tightbinding model used for TiSe_{2} in ref. ^{15}. Accordingly, we defined the orbital basis in the octahedral coordinates {\(x^{\prime} y^{\prime} z^{\prime}\)} in Fig. 4d, considering the three Te 5p orbitals p {\({p}_{x}^{\prime},{p}_{y}^{\prime},{p}_{z}^{\prime}\)} for each of the two chalcogen sites (Te(1) and Te(2)), and the five Ti 3d orbitals dγ {(\({d}_{x{^{\prime} }^{2}y{^{\prime} }^{2}},{d}_{3z{^{\prime} }^{2}r{^{\prime} }^{2}}\))} and dϵ {(\({d}_{x^{\prime} y^{\prime} },{d}_{y^{\prime} z^{\prime} },{d}_{x^{\prime} z^{\prime} }\))}. The hopping parameters and onsite energies reported in Table 1 were determined by initially fitting the tightbinding model along MΓK to the DFT band structure, and subsequently refining the fits using our experimental ARPES data of the undistorted phase (see Supplementary Fig. 8). An onsite spinorbit coupling term on the Te sites is included (λ = 0.38 eV), optimised from fitting the dispersion on the high temperature ARPES spectrum.
Overlap evaluation
The orbital overlaps in Fig. 4c were calculated considering the tesseral harmonics of the six Te p orbitals at the vertex of the TiTe_{6} octahedron defined in the global coordinate system, {x, y, z} and the Ti \({d}_{x^{\prime} y^{\prime} }\) orbital at the centre of the TiTe_{6} octahedron defined in the octahedral coordinate system \(\{x^{\prime} y^{\prime} z^{\prime} \}\). The overlap integral was evaluated numerically on a (121 × 121 × 121) cubic mesh. We simulate the effect of the phonon mode softening by displacing the atoms along the direction indicated by the black arrows in Fig. 4c, following the distortion pattern reported for TiSe_{2} in ref. ^{7}. Since Ti is lighter that Te, we assume the Ti displacement is 9 times larger that the Te ones.
Minimal model
As basis for the Hamiltonian in Eq. (1) we choose the five bands involved in the hybridisation near the Fermi level. The three electron pockets (e_{i}, i = 1, 2, 3) are parametrised as 2D elliptic paraboloids, while the two hole bands as circular nonparabolic bands:
where e_{h} and e_{c} are the band minimum energy for the valence and conduction band respectively; μ and ν are the effective masses along the major and minor axis of the elliptical parabola, and α_{in} and α_{out} are the nonparabolic terms for the inner and outer valence band following the Kane model^{46}. All these coefficients were determined by fitting the simulated spectrum with Δ = 0 on the experimental data at T = 160 K shown in Fig. 5c.
In order to extend the model to the bulk case we introduced and additional cosine dispersion in k_{z} for the onsite energies:
where the values of e_{c} at M and L and e_{h} at Γ and A were determined by fitting the ARPES data in Fig. 6a. To ensure charge neutrality, a shift of the chemical potential across the CDW transition was taken into account, simulating how the occupied DOS depends on the hybridisation strength (Δ) at constant T = 16 K (see Supplementary Fig. 9 for details).
ARPES simulation
The ARPES simulations in Fig. 5c, e were performed taking into account the intensities of the 5 bands (e_{1}, e_{2}, e_{3}, h_{in}, h_{out}) considered in the minimal model plus a background with thirdorder polynomial along the momentum k and a linear dispersion along the energy axis ω. Each band intensity I_{B}(k, ω) takes the form of:
where M is the matrix element, A is the spectral function, f(T, ω) is the Fermi Dirac distribution at temperature T. The entire expression is then convolved with a 2D Gaussian (R) to simulate the experimental energy and momentum resolution. M depends on the band character C_{b} and can be approximated to have a linear dependence on k:
A takes into account the Lorentzian broadening of the band intensity:
where e_{b} is the bare band dispersion calculated by the minimal model and Σ_{b} is the Lorentzian linewidth which includes both impurity scattering and an effective broadening due to the rotational disorder discussed in Supplementary Fig. 7.
For measurements performed at the Brillouin zone centre, the matrix element can be expected to approximately follow that of the original valence band character^{47,48} (dark blue in Fig. 5b, d). We note that for both the high and low T spectra the band intensities decrease monotonically from −0.16 eV up to the Fermi level. Since it is not a temperaturedependent feature we do not consider it related to the CDW transition, but rather is likely a matrix element variation. In order to have a better agreement with the experimental data, we include a temperatureindependent intensity decay, I_{dec}, of the form:
where the coefficients E_{f} and E_{i} were determined fitting the hightemperature data and kept constant for all the simulations at different temperatures. Thus, this contribution is cancelled out when we calculate the intensity difference between the high and lowtemperature data as in Fig. 3.
Data availability
The research data supporting this publication can be accessed at https://doi.org/10.17630/b2bb58a28439434ca7a609a0a2f68947^{49}.
Code availability
The codes that were used here are available upon request to the corresponding author.
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Acknowledgements
We thank Sebastian Buchberger, Brendan Edwards, Lewis Hart, Chris Hooley, Federico Mazzola, Martin McClaren, Philip Murgatroyd, Luke Rhode and Gesa Siemann for useful discussions and technical assistance. We gratefully acknowledge support from the Leverhulme Trust and the Royal Society. Via membership of the UK’s HEC Materials Chemistry Consortium, which is funded by the EPSRC (EP/L000202, EP/R029431, EP/T022213), this work used the ARCHER2 UK National Supercomputing Service (www.archer2.ac.uk) and the UK Materials and Molecular Modelling (MMM) Hub (Thomas  EP/P020194 & Young  EP/T022213). W.R. is grateful to University College London for awarding a Graduate Research Scholarship and an Overseas Research Scholarship. O.J.C. and K.U. acknowledge PhD studentship support from the UK Engineering and Physical Sciences Research Council (EPSRC, Grant Nos. EP/K503162/1 and EP/L015110/1). I.M. and E.A.M. acknowledge studentship support from the International MaxPlanck Research School for Chemistry and Physics of Quantum Materials. S.R.K. acknowledges the EPSRC Centre for Doctoral Training in the Advanced Characterisation of Materials (CDTACM, EP/S023259/1) for funding a PhD studentship. The MBE growth facility was funded through an EPSRC strategic equipment grant: EP/M023958/1. We thank SOLEIL synchrotron for access to the CASSIOPEE beamline (proposal Nos. 20181599 and 20171202). The research leading to this result has been supported by the project CALIPSOplus under Grant Agreement 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020.
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M.D.W., A.R., O.J.C., K.U., I.M., E.A.M. and P.D.C.K. measured the ARPES data; T.A. performed the data analysis; W.R. and S.R.K. performed the DFT calculations, which were analysed by W.R., S.R.K., T.A., and D.O.S; T.A. performed the tightbinding calculations, ARPES simulations and matrix element calculations; M.W., T.A. and P.D.C.K. developed the minimal model; A.R. and A.D. grew the monolayer samples; K.R. grew the single crystal samples; P.L.F. and F.B. maintained the CASSIOPEE beam line and provided experimental support; P.D.C.K. led the project; T.A. and P.D.C.K. wrote the manuscript with contributions from all authors.
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Antonelli, T., Rahim, W., Watson, M.D. et al. Orbitalselective band hybridisation at the charge density wave transition in monolayer TiTe_{2}. npj Quantum Mater. 7, 98 (2022). https://doi.org/10.1038/s41535022005089
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DOI: https://doi.org/10.1038/s41535022005089
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