Orbital- and $k_z$-Selective Hybridization of Se $4p$ and Ti $3d$ States in the Charge Density Wave Phase of ${\mathrm{TiSe}}_2$

We revisit the enduring problem of the $2\times 2\times 2$ charge density wave (CDW) order in ${\mathrm{TiSe}}_2$, utilizing photon energy-dependent angle-resolved photoemission spectroscopy to probe the full three-dimensional high- and low-temperature electronic structure. Our measurements demonstrate how a mismatch of dimensionality between the 3D conduction bands and the quasi-2D valence bands in this system leads to a hybridization that is strongly $k_z$ dependent. While such a momentum-selective coupling can provide the energy gain required to form the CDW, we show how additional “passenger” states remain, which couple only weakly to the CDW and thus dominate the low-energy physics in the ordered phase of ${\mathrm{TiSe}}_2$.

Figure 1

(a) Valence band dispersion along $A\text{−}L$ at $T=300\mathrm{K}$, measured at a photon energy $h\nu =121\mathrm{eV}$, in $p$ polarization ($p$ and $s$ polarizations correspond to linear horizontal and linear vertical polarizations, respectively, and the analyzer slit is aligned vertically at the I05 beam line). (b) Constant energy map at $E_F$ (Fermi surface) at $T=250\mathrm{K}$ ($h\nu =121\mathrm{eV}$, $p$ polarization). (c) Fermi surface of ${\mathrm{TiSe}}_2$ as calculated with a standard DFT code [27]. (d) $k_z$ map, processed from photon-energy dependent data at $E-E_F=-380\mathrm{meV}$. The inset shows the conduction band at $E-E_F=+20\mathrm{meV}$ from the same data set. (e)–(g) Dispersions at $\mathrm{T}=300\mathrm{K}$ acquired in $s$ polarization with $h\nu =44$, 95, and 119 eV, approximately corresponding to $L\text{−}A\text{−}L$, $M\text{−}\mathrm{\Gamma }\text{−}M$, and $L\text{−}A\text{−}L$ paths, respectively. (h) Comparison of EDCs at high-symmetry points, indicating an indirect band gap of $74\pm 15\mathrm{meV}$. The data in panels (e)–(h) are divided by the Fermi function to reveal the spectral weight above $E_F$.

Figure 2

(a) Schematic $k_z$ dispersions in the normal state. The conduction bands are plotted with a displacement of $q_{\mathrm{CDW}}$. The $4p_z$ valence band is not included. (b) Schematic $k_z$ dispersions in the 3Q CDW phase, showing $k_z$-selective band hybridization. We do not include the backfolded bands. (c) Tight-binding calculations of monolayer ${\mathrm{TiSe}}_2$ showing orbital-selective band hybridization in the CDW phase. Dashed gray lines represent band dispersions at infinitesimal distortion. Color and weight of lines project the difference between the Ti $3d_{z^2}$ character and other $3d$ components (see SM [23] for further plots and details). (d) Experimental $k_z$ dispersion along $M\text{−}L\text{−}M$ in the CDW phase. The intensity is dominated by bright backfolded valence bands, which display characteristic dips around $L$ points. Conduction pockets are also observed, centered at $L$ points.

Figure 3

(a) Valence band dispersion along $A\text{−}L$ at 10 K ($h\nu =121\mathrm{eV}$, $p$ polarization). (b) Fermi surface map, showing near-circular pockets at the $L$ points. (c) In-plane construction of the 3Q ordered phase; backfolded bands are dashed, the reconstructed low-temperature Fermi surface is shown inset. (d) Low-temperature $k_z$ map, equivalent to Fig. 1, now showing backfolded valence bands, and the Fermi surface in the inset (top right). (e),(f) High symmetry dispersions along $\mathrm{\Gamma }\text{−}M$ (31, 100 eV) and $A\text{−}L$ (44, 121 eV, all $s$ polarization), respectively. (g) Schematic band structure at ${\mathrm{\Gamma }}^{*}$ of the $2\times 2\times 2$ Brillouin zone, with an estimated band gap of $\sim 15\mathrm{meV}$.