Excitons and Optical Response in Excitonic Insulator Candidate TiSe$_2$

Abstract

The origin of the charge density wave (CDW) phase in TiSe$_2$ is a highly debated topic, with lattice and excitonic correlations proposed as the main driving mechanisms. One of the proposed scenarios is the excitonic insulator (EI) mechanism, where soft electronic mode drives the phase transition. However, the existence of this purely electronic mode is controversial. Here, we perform fully ab-initio calculations of the electron excitation spectra in TiSe$_2$ with electron-hole excitonic effects included via Bethe-Salpeter equations. In the normal high-temperature phase the excitation spectra is dominated by the exciton mode at 1.6 eV, while no well-defined soft electronic modes that could support the EI phase are present. In the CDW phase, the structural distortions induce a CDW band-gap opening between Ti-$d$ and Se-$p$ states, which supports the formation of the two low-energy excitonic modes in the optical spectrum at 0.4 eV and 80 meV. Close to the transition temperature $T_{\rm CDW}$, these two excitonic modes are softened and approach zero energy. These results suggest that the EI mechanism is not a main driving force in the formation of the CDW phase in TiSe$_2$, but there is a region in the phase diagram near $T_{\rm CDW}$ where EI fluctuations could be relevant.

Figures (2)

• Figure 1: (a) Crystal structure of TiSe$_2$ in the normal high-temperature phase. (b) Electronic band structure of TiSe$_2$ along high-symmetry points as calculated with DFT-PBE functional. The orange and blue arrows depict the vertical optical ($\mathbf{q}=0$) and finite-momentum low-energy ($\mathbf{q}=\mathrm{M}$) electron-hole transitions. (c) Optical absorption spectrum of TiSe$_2$ as calculated with BSE (that includes electron-hole interaction) and independent-particle RPA (no electron-hole interaction) approximations shown with blue and red dashed lines, respectively. These calculations were done on top of PBE electronic structure. (d) Absorption spectra obtained with BSE and RPA methods for three different momenta $\mathbf{q}$ close to the M point of the BZ. The topmost spectrum is calculated for $\mathbf{q}=\mathrm{M}$.
• Figure 2: (a) Electronic band structure of TiSe$_2$ in the CDW phase at 100 K as obtained with PBE and HSE functionals. The CDW transition temperatures as obtained with PBE and HSE are 1100 K and 2000 K, respectively. The Se-$p$ states are denoted with v$_1$, while three backfolded Ti-$d$ states are labeled with c$_1$ and c$_2$ (note that the latter two are almost degenerate). Orange and red arrow show the electron-hole transitions that lead to the formation of low-energy excitons. (b) The energy of the CDW gap at the $\Gamma^{\ast}$ point and PLDs of Ti atoms as a function of temperature obtained with the PBE functional. Inset shows the $2\times 2$ CDW cell with directions of the PLDs. (c) Optical absorption spectra for the CDW phase at 100 K as obtained with HSE-RPA and HSE-BSE methods. The experimental data is from Ref. li2007semimetal. (d) The HSE-BSE optical absorption spectra as a function of temperature across the CDW transition. Three excitonic peaks are labeled with arrows. (e) The energy peaks of the three excitonic modes as a function of temperature. The full and open blue squares are obtained with two different scissor corrections as explained in the text. The data are compared with optical absorption li2007semimetal and RIXS monney12 results. The blue shaded area denotes the CDW phase dominated by the electron-phonon coupling, while the orange area approximately marks the range where EI fluctuations might be important. (f) The intensity of absorption at $\omega_{2}^{\rm exc}$ as a function of temperature and normalized to the value at the lowest temperature.