Observation of orbital order in the van der Waals material $1T\text{−}{\mathrm{TiSe}}_2$

Besides magnetic and charge order, regular arrangements of orbital occupation constitute a fundamental order parameter of condensed matter physics. Even though orbital order is difficult to identify directly in experiments, its presence was firmly established in a number of strongly correlated, three-dimensional Mott insulators. Here, reporting resonant x-ray-scattering experiments on the layered van der Waals compound $1T\text{−}{\mathrm{TiSe}}_2$, we establish that the known charge density wave in this weakly correlated, quasi-two-dimensional material corresponds to an orbital ordered phase. Our experimental scattering results are consistent with first-principles calculations that bring to the fore a generic mechanism of close interplay between charge redistribution, lattice displacements, and orbital order. It demonstrates the essential role that orbital degrees of freedom play in ${\mathrm{TiSe}}_2$, and their importance throughout the family of correlated van der Waals materials.

Figure 1

Charge and orbital order in $1T\text{−}{\mathrm{TiSe}}_2$. (a) Top-view and three-dimensional sketch of the atomic structure within a single layer of $1T\text{−}{\mathrm{TiSe}}_2$ in the high-temperature phase (space group 164, $P\overline{3}m1$). (b) The charge ordered phase without relative phase shifts (space group 165, $P\overline{3}c1$). The local charge transfer process for one of the CDW components and all orbitals involved in it are shown in the three-dimensional structure, followed by a top view of the displacements and charge transfers in a single CDW component, and finally the full displacement pattern in the triple-$Q$ structure actually realized in ${\mathrm{TiSe}}_2$. Extrema of the CDW are indicated by faint gray lines and atomic displacements by arrows (exaggerated for clarity). Double lines indicate an electronic bonding state between neighboring atoms, while dashed lines depict an electronic antibonding state. (c) The orbital ordered state proposed in Ref. [26], resulting from relative phase shifts $\delta$ between CDW components (space group 5, $C2$). The sliding of a single component causes the charge transfer processes to be centered on either the upper or lower Se layer. As a result, the atomic displacements and orbital occupations associated with different CDW components differ in amplitude. This results in a slight alteration to the distortion pattern, as well as an ordered redistribution of electronic charge among the titanium $t_{2g}$ orbitals.

Figure 2

Charge density wave and orbital order in ${\mathrm{TiSe}}_2$ and their energy and temperature dependence measured by RXS at the Ti K edge. (a, b) H cuts at 11 K for $(0.5,0.5,0.5)$ and $(0.5,0,0.5)$, respectively. (c, d) Corresponding energy scans near the Ti K edge. The x-ray-absorption spectrum (XAS) at the Ti K edge is shown for comparison, as is the energy scan for the $(0,0,1)$ Bragg peak. (e, f) Temperature dependence at 4.969 keV of the peaks at $(0.5,0.5,0.5)$ and $(0.5,0,0.5)$, respectively. The resistivity measurement is displayed in (e) for comparison, with the inset showing its derivative close to the transition temperature. The gray solid lines are fits to the data showing a transition temperature $T_{\text{cdw}}\approx 193\mathrm{K}$.

Figure 3

Energy dependence of the scattering amplitude in the orbital ordered phase. (a) Expected energy dependence of the scattering amplitude, with its characteristic resonance structure, based on the square of the orbital-resolved projected densities of states (see Supplemental Material [50]), and broadened to reflect the effect of finite lifetimes. Different curves represent different values of the relative phase shift between CDW components. The energy scale on the horizontal axis is measured with respect to the Fermi level. (b) Ti $4p$ orbital densities at fixed energy in the atomic structures with zero (left) and nonzero (right) relative phase shift $\delta$ between CDW components. For $\delta =0$, three out of four Ti sites have one predominant $4p$ orbital (green), while orbitals with lower density are degenerate. The structures in neighboring planes are related by inversion. For nonzero relative phase shift, the densities of some previously degenerate orbitals increase (green) or decrease (red). The pattern of orbital polarization shown here for Ti $4p$ orbitals is also present in the densities of other orbitals, and breaks both inversion and threefold rotational symmetry.