Correlation between electronic and structural orders in 1T-TiSe2

The correlation between electronic and crystal structures of 1T-TiSe2 in the charge density wave (CDW) state is studied by x-ray diffraction. Three families of reflections are used to probe atomic displacements and the orbital asymmetry in Se. Two distinct onset temperatures are found, TCDW and a lower T* indicative for an onset of Se out-of-plane atomic displacements. T* coincides with a DC resistivity maximum and the onset of the proposed gyrotropic electronic structure. However, no indication for chirality is found. The relation between the atomic displacements and the transport properties is discussed in terms of Ti 3d and Se 4p states that only weakly couple to the CDW order.


3
DC resistivity shows an increase at TCDW which could be due to charge localization and gap formation, similar to other CDW systems [25][26][27]. However, despite the reconstruction of the band structure at TCDW, a large and broad peak in the resistivity is seen at a temperature lower than TCDW [T* » 165 K, see Fig. 2(a)]. It has been suggested that this anomalous peak originates from a crossover between two temperature regimes: from the low-temperature state with negative carriers (electrons) resulting from a small extrinsic doping to a high-temperature state with both negative and positive carriers (electrons and holes) which arise due to thermal fluctuations [21].
Besides the anomalous resistivity peak, the CDW state also features a circular photogalvanic current interpreted as the occurrence of gyrotropic electronic order [5] and the emergence of chirality [28], both of which have distinct onset temperatures below TCDW. The chiral phase transition, as claimed to be observed by scanning tunneling microscopy (STM) and x-ray diffraction (XRD) experiments [28][29][30], takes place » 7 K lower than TCDW [28].
Such a chiral CDW order with P1 symmetry is manifested by chiral atomic displacements and a possible orbital order [31][32][33][34][35]. However, these experimental findings have been challenged by alternative interpretations [36][37][38]. It is, therefore, important to clarify the electronic and crystal structures and their temperature dependence.
In this Letter, we study in detail the various structural modifications below TCDW in a thin flake of single crystalline 1T-TiSe2. The Bragg reflections, which correlate to the CDW order, are ascribed to specific atomic motions. The change in transport properties below TCDW is discussed in terms of a possible change of orbital hybridization caused by these atomic motions. Although the onset temperature of certain reflections deviate from TCDW, no evidence of chirality [28] is found. Detailed investigations of a space-group forbidden reflection in resonant XRD combined with ab initio simulation are well described by a non-chiral symmetry of electronic and crystal structures in the CDW state.
Figures 1(a) and 1(b) display the crystal structure of 1T-TiSe2 in the CDW state (P3 " c1) [22]. Both Ti and Se atoms, each of which occupies a single crystallographic site in the normal state, become two distinct sites in the CDW state. While Ti1 and Se2 have only single in-plane (δTi) or out-of-plane (δSe2_out) displacements from the normal state, respectively, Se1 has both types of components (δSe1_in and δSe1_out in-plane and out-of-plane, respectively). Since there are two Se sites, the out-of-plane motions of these sites are independent and we can define their difference as ΔzSe = δSe1_out -δSe2_out. 4 A single crystal of 1T-TiSe2 grown by the iodine vapor transport method [24] was cleaved along the (001) plane by repetitive exfoliation, and a flake (» 2 µm thickness) was mounted on a polycrystalline diamond substrate. Experiments were performed at the beamline I16 [39] of Diamond Light Source. The sample was mounted on the cold finger of a closed cycle refrigerator attached to a Newport six-circle kappa diffractometer and cooled down well below TCDW and T*. The photon energies of x-ray beams were tuned to 12.6 keV for nonresonant XRD and around the Se K edge (» 12.658 keV) for resonant XRD. A Si (111) analyzer was used to determine the polarization state of the scattered beam. Scattered photons were counted using a PILATUS 100K pixel detector [40] and using an avalanche photo diode during azimuthal scans at resonance. A rocking curve as well a raw two-dimensional image of (1 0 7) shown in Fig. S2 confirm the high crystalline quality of the sample. X-ray absorption spectrum (XAS) was obtained by integrating the fluorescence signal. DC resistivity was measured for a single crystal from the same batch with that used for this XRD study along the (001) plane by the four-points resistance method.
We classify the measured Bragg reflections into three families: A, B and C, all of which are space-group forbidden in the normal state. We denote reflections by using the indices in the normal state, h, k and l. Family A represents (h k l) reflections where l and at least one of h or k are half-integers. These reflections appear at G + q, where G is a reciprocal lattice vector of the normal state, and thus directly measure the displacement of the lattice from the CDW order. Family B represents (0 k l) type of reflections where l is an integer while k is a halfinteger. Family C also represents (0 k l) type of reflections but with both l and k as half-integers.
Note that family C are space-group forbidden even in the CDW state and are observed only at resonance (as shown later). Figure 2 shows the temperature dependence of the integrated intensities of the three families together with the DC resistivity. The reflections (0.5 1 1.5) (family A, off-resonance) and (0 0.5 9.5) (family C, at resonance) appear at TCDW and their intensities grow proportional to the square root of TCDW -T. On the other hand, the (0 0.5 8) (family B, off resonance) appears at T*, which coincides with the DC resistivity maximum, and shows an approximately linear dependence with temperature (see Fig. S5 for further reflections). These differences imply that A and B are sensitive to different atomic displacements allowed in the CDW state and that the displacements that dominates B have a correlation to the transport properties. The previously reported difference in the onset temperatures (» 7 K) of (1.5 1.5 0.5) and (2.5 1 0) [28], as well as the linear temperature dependence of (2.5 1 0) are reasonably explained by a fourth-order contribution of the in-plane CDW distortion occurring at TCDW [33]. However, the observed difference in the onset temperatures (» 37 K) in our study cannot be explained in the same way since a calculated fourth-order contribution to the intensity [black curve in Fig. 2

(a)]
does not fit to the experimentally obtained data for (0 0.5 8).
The (0 0.5 9.5) reflection, which is space-group forbidden in the CDW state, is observed only around the Se K edge as seen in Fig. 3(a). The intensity depends on the azimuthal angle, and the profile is nicely described by ab initio calculations performed by the FDMNES code using a model based on the crystal structure in the CDW state [22] [see Fig. 3 The x-ray absorption at the Se K edge includes the excitation of an electron from the 1s state to the 4p valence state, which would be totally occupied in a fully ionic picture. Thus, the observation of the space-group forbidden reflection means that there are holes in the Se valence state in addition to the aspheric electron distribution in the Se1 4p, as predicted in Ref. 43 by DFT calculations. It is obvious that the change in orbital asphericity is associated with the CDW order since its onset temperature coincides with the appearance of reflections of family A. The absence of calculated intensities for artificially removed in-plane CDW distortions for the reflection further confirms this relation [41]. In addition, the anisotropic environment of the hybridized Ti 3d states, associated with the CDW structure, allows the formation of an excitonic state [3,4] generating holes in the Se 4p bands which influence the electron distribution in Se.
To identify the nature of atomic displacements dominating space-group allowed reflections A and B, numerical calculations for possible models with different types of displacements were carried out and are summarized in Table S2 (see Ref. 41 for details of the models and procedure). We find that δTi and δSe1_in, both in-plane displacements, dominate A while ΔzSe, relating to an out-of-plane displacement, dominates B. Figure 4 shows the largely dominated by δTi through a quadratic contribution to the intensity (linear in the structure factor) without any contribution from ΔzSe. On the other hand, B is dominated by ΔzSe (µ ΔzSe 2 ) with a 10 3 times smaller quartic contribution in δTi. The quartic contribution of δTi to the intensity can explain the linear temperature dependence of B but cannot explain the significant intensity and the difference in onset temperature from A. Thus, it is plausible that ΔzSe sets in » 37 K lower than TCDW and develops slower as a function of temperature while cooling than the CDW order parameters, δTi and δSe1_in. These observations are possible when ΔzSe is not the primary order parameter of the CDW phase transition. The difference in onset temperature between (1.5 1.5 0.5) and (2.5 1 0) reported earlier [28,33] are not related to our observations since (2.5 1 0) is independent of ΔzSe (as l = 0). As shown in Ref. 33, the temperature dependence is naturally explained by the quartic behavior of the in-plane displacements.
However, the anomaly in the specific heat found in that study [28] might be directly related to ΔzSe.
Whereas the CDW order largely affects the band structure, the maximum of the DC resistivity is not correlated to TCDW but T*, around which the gyrotropic electronic order also appears [5]. Note that TCDW can vary between different samples, probably due to differences in defect density [1]. The displacements δTi and δSe1_in dominate the CDW distortion and ΔzSe does not further break the symmetry in the CDW state. The ΔzSe, the magnitude of which is much smaller than δTi and δSe1_in, creates only a tiny change in the structure of the main bands that form the hybridization gap. The onset of δSe1_in shortens the Se1-Ti bond length compared to Se2-Ti and, hence, Se1 moves away from the adjacent Ti layer below T*. This weakens the orbital hybridization between Ti 3dz2 and Se 4px,y at the A point, increasing the band curvatures of the specific branches [compare Figs. 1(c) and 1(d)]. Therefore, the carriers in Ti 3dz2 and Aderived Se 4px,y acquire lighter effective masses and longer mean free paths, resulting in a decrease of DC resistivity below T*. Note that the Ti 3dz2, which crosses the Fermi energy, dominates the transport in the CDW state.
Interestingly, for both CuxTiSe2 (x = 0.05) and 1T-TiSe2 at ³ 2.82 GPa, the anomalous peak in the DC resistivity vanishes and a superconducting state appears upon lowering the temperature [6,7]. These samples have the same space group and CDW transition as 1T-TiSe2 under ambient pressures, but the distortion ΔzSe is zero in the CDW state for both the doped and pressurized samples [22]. This implies that ΔzSe plays a critical role in the transport properties and might be a key parameter governing the emergence of the superconducting state. 7 We now comment on the possible occurrence of chirality in the CDW state. Even though the B reflections have a lower onset temperature than TCDW, as there is no indication for a first order transition, a symmetry analysis based on the Landau theory is inconsistent with the occurrence of chirality [5]. The circular photogalvanic current, which is claimed as the evidence for a chiral electronic structure, appears at almost the same temperature as the structural modification [5]. This suggests that chirality in the electronic states is significantly coupled to the lattice, and the underlying crystal lattice should be chiral. Observation of orbital asphericity through studying the polarization dependence of space-group forbidden reflections in resonant XRD is one of the established methods to determine the presence of chirality in crystal lattices [44][45][46]. However, the observed azimuthal-angle dependence of the (0 0.5 9.5) reflection is very well reproduced by the ab initio calculations assuming the non-chiral space group P3 " c1. Moreover, this reflection must be a symmetry allowed reflection if the space group is P1, as claimed in earlier studies [31][32][33][34][35]. The absence of an intensity for this reflection at off-resonance and in the σ-σ¢ channel is shown in Figs. 3(a) and 3(b), respectively. Therefore, we can conclude that our results support neither the chirality of the Se orbitals nor the low symmetry crystal structure in the CDW state.
In summary, we examined the charge density wave (CDW) phase of 1T-TiSe2 by means of x-ray diffraction. One of the studied three families of reflections has different onset temperature compared to the others. Our detailed analysis revealed that its origin comes from the difference in relative out-of-plane motion between the two Se sites, which weakly couples to the CDW order. The different onset temperature compared to the CDW transition of the reflection is not caused by the onset of chirality. Such out-of-plane atomic displacements of Se can reduce the orbital hybridization between Ti 3dz2 and Se 4px,y states, which couples only weakly to the CDW order. The consequent decrease of the effective mass of carriers results in a reduction of DC resistivity at lower temperatures. Our x-ray diffraction results provide the crucial link between the origin of the anomalous feature in transport properties and structural modifications that change the hybridization in the relevant electronic bands.    These data were taken at the azimuthal angle indicated by a gray dotted line in Fig. 3(b). 19

I.
Symmetry analyses and calculations for the (0 0.5 9.5) reflection (family C) In order to clarify the origin of the space-group forbidden (0 0.5 9.5) reflection (family C), we show our symmetry analysis and results of ab initio and DFT calculations. In general, due to intra-atomic excitation of an electron between specific orbitals in the excited atoms, resonant x-ray diffraction is sensitive to the anisotropic electron distribution or orbital asphericity. This anisotropic scattering is described by an x-ray susceptibility tensor $ , which is defined by the global and local symmetries of the atoms [2], and leads to the emergence of a space-group forbidden reflection. We perform the symmetry analyses for Se1 and Se2, respectively, as different crystallographic sites give space-group allowed reflections while our discussion is dedicated to reveal the origin of the space-group forbidden reflection.
The local C3 symmetry requires the relation, $ = 3 $ ! "# , so that one finds $$ = %% and $% = %& = $& = 0 , meaning that the symmetry-adopted $ possesses only diagonal components Here N is the number of unit cells in the crystal. Thereby, Se2 is found to be independent of the (0 0.5 9.5) reflection. This is cross-checked from the result of ab initio calculation done by using the FDMNES code [3] and simply removing Se1 from the reported crystal structure in the CDW state [4] [see Fig. S4(a)].
The low-symmetry Se1 site locates at the Wyckoff position 12g, which is the general position of space group P3 " c1 and has the local symmetry 1. Thus, all the components in $ are symmetry adapted. Positions, symmetry relations, phase factors in . and $ are given for all Se1 in the unit cell in Table S1. From the parameters shown in the table, one finds that . is denoted as From Eqs. (4)- (7), . is found to possess only the xy (Fxy) and xz (Fxz) components and is represented as Hereby we use the Cartesian coordinate ξηζ defined by the diffraction geometry; η is normal to the scattering plane, ζ is along Q and ξ is normal to both the directions [see Fig. S4 where we set [100] along the scattering plane. The azimuthal angle ψ is defined along Q, and ψ dependence of 012 I is given as and is decomposed at the respective channels, From Eqs. (11) and (12), one finds the intensity at the channels as The experimental data as well the results of ab initio calculations shown in Fig. 3(b) are reproduced by Eqs. (13) and (14) [see Fig. S4(c)]. Furthermore, ab initio calculation with removing Se2 from the crystal structure in the CDW state [4] shown in Fig. S4(a) cross-checks the results of our symmetry analysis. Therefore, Se1 is found to be responsible to the spacegroup forbidden (0 0.5 9.5) reflection. We remark that the (0 0.5 9.5) reflection is space-group forbidden and, thereby, the scattering for the reflection is not caused by a charge but an aspheric orbital (or a quadrupole moment), whose orientation is modulated in the unit cell.  By removing all the in-plane displacements appearing at the CDW order, i.e., for both Ti1 and Se1, we perform the ab initio calculations for the (0 0.5 9.5) reflection but no significant intensity is obtained at both the channels [see Fig. S4(b)]. The results mean that the in-plane displacements cause an aspheric and spatially modulated orbital state at Se1, which is responsible for the (0 0.5 9.5) reflection. This is supported by the onset temperature of reflections of family C that is the same as that of A (at TCDW) as shown in Fig. 2.
Visualization of the electronic density distribution of states involved to the CDW transition yields an intuitive picture of the asphericity of the orbitals [5]. Figure S4 Fig. 1(a).
(e) Diffraction geometry and Cartesian coordinate systems used in the symmetry analyses.

II. Characterization of further reflections
In addition to the (0.5 1 1.5) and (0 0.5 8) reflections, which are shown in the main text, we characterized further reflections. Another flake of a single crystal of 1T-TiSe2 was measured at the X04SA beamline of Swiss Light Source [7]. The sample was cooled down well below TCDW and T* by using a N2 cryoblower. The photon energy of x-ray beams was tuned to 9.2 keV, and signals were detected by a PILATUS II photon-counting pixel detector [8].

III. Numerical calculations of diffraction intensity
The calculations of diffraction intensities for the space-group allowed reflections, A At first, it is clear from Fig. S6(e) that the in-plane off-axes displacements show tiny contribution for any reflections than the others. The reflections of family A are dominated by the in-plane displacements of Ti and Se along the hexagonal axes (µ δTi 2 , δSe1_in 2 ) without any contribution from the relative out-of-plane displacements of two Se as seen in Table S2. On the other hand, except the (1.5 0 2) reflection, which has a very weak diffraction intensity, the reflections of family B are basically dominated by the out-of-plane displacements (µ ΔzSe 2 ) with 10 3 or 10 4 times smaller quartic contribution from the in-plane displacements (µ δTi 4 , δSe1_in 4 ). The (2.5 1 0) reflection, which has a different onset temperature compared to (1.5 1.5 0.5) on first inspection and was discussed as the indication of a chiral phase transition [10], is not classified into any of the three families and is dominated by the in-plane displacements with a quartic contribution (µ δTi 4 , δSe1_in 4 ) as reported in Ref. 11.   [4]. The family C reflection is space-group forbidden in the CDW state.