Excitonic insulator emerging from semiconducting normal state in $1T\text{−}{\mathrm{TiSe}}_2$

A new state of matter, an excitonic insulator (EI) state, was predicted to emerge from Bose-Einstein condensation of electron-hole pairs. Some candidate materials were suggested but it has been elusive to confirm its existence. Recent works gave renewed support for the EI picture of the charge density wave (CDW) state below the critical temperature $T_c\approx 200$ K of $1T\text{−}{\mathrm{TiSe}}_2$. Yet, an important link to its establishment is to show that a majority fraction of the measured $T_c$ indeed follows from the Coulomb interaction alone, while a quantitative match of the $T_c$ may require assistance from the electron-lattice coupling. This will establish that the CDW is formed predominantly by the Coulomb interaction and help confirm the EI view for ${\mathrm{TiSe}}_2$. Here, we provide such calculations by solving the exciton gap equation with material-specific electronic structures. We obtain, with no fitting parameters, $T_c\approx 135\pm 27$ K for the normal state gap of $E_g\approx 74\pm 15$ meV. It seems that the calculated $T_c$ from Coulomb interaction gives a majority fraction of experimental $T_c$ for recently determined values of $E_g$. The measured doping dependence of $T_c$ was satisfactorily reproduced as well. Also in agreement with experiments is the same set of calculations of the photoemission spectroscopy and density of states. The semiconducting state above and EI below $T_c$ together should give a coherent picture of $1T\text{−}{\mathrm{TiSe}}_2$.

Figure 1

(a) The first BZ of $1T\text{−}{\mathrm{TiSe}}_2$ exhibiting the hole band (blue) around the $\mathrm{\Gamma }$ and three electron bands (red) around $M_i$ points. The BZ is reduced in the charge density wave (CDW) phase (black dashed line) and appear as the backfolded dispersions (red and blue dashed lines). (b) The energy dispersion of the hole and electron bands along the $\mathrm{\Gamma }\text{−}{M}_1$ direction. $E_g$ is the indirect bare gap between the electron and hole bands. The dashed lines are backfolded bands.

Figure 2

(a) The EI phase diagram of $1T\text{−}{\mathrm{TiSe}}_2$ exhibiting $T_c$ as a function of $E_g$ for ${\epsilon }_{\infty }=3.0$. The grey dashed line indicates the border between semimetal and semiconductor phases. The red star on the $T_c$ curve shows the calculated $T_c=135$ K corresponding to the recent measurement of $E_g=74$ meV [18]. The inset shows an enlargement of the region around the red star. (b) The calculated doping dependence of ${\mathit{T}}_c$ for $E_g=0.12$ eV in comparison with experiments. The red circles represent measurements on pristine and Cu intercalated polycrystals [19].

Figure 3

The momentum dependence of the computed order parameter ${\mathrm{\Delta }}_i\left(\mathbf{k}\right)$ within the first BZ. The blue and red lines are the hole and electron bands, respectively. ${\mathrm{\Delta }}_i\left(\mathbf{k}\right)$ takes the largest value at the BZ center and is elongated along $\mathrm{\Gamma }\text{−}{M}_i$ direction.

Figure 4

(a, b) The calculated spectral function around $\mathrm{\Gamma }$ and $M$ points. The orange dashed lines represent the (a) hole and (b) electron dispersions in the normal phase and the black dashed lines show their backfolded ones. The high-intensity curves near the orange lines are the main dispersions in the EI state. (c) DOS in EI (black line) and normal (red) states. The black and green arrows indicate the coherence peaks due to flat dispersions of spectral function. These peaks are split because of the momentum dependence of ${\mathrm{\Delta }}_i\left(\mathbf{k}\right)$. (d) The normalized DOS, where the zero-bias peak (marked by the red arrow) shows up because of the normal state gap.

Figure 5

The gray hexagonal area represents 1st BZ of ${\mathrm{TiSe}}_2$. It can be converted into the blue dash rhombus of the same area. The red and black dots in the blue dashed rhombus are the selected $k$ points for FFT. The red dots are the $k$ points in the irreducible BZ (pale red triangle) for $T_c$ calculations.

Figure 6

(a) The critical temperature $T_c$ of excitonic instability as a function of the energy gap for four different characteristic configurations of electronic structure. (b) $T_c$ vs $E_g$ for ${\mathrm{TiSe}}_2$ for different choices of the background dielectric constant ${\epsilon }_{\infty }$.

Figure 7

Contour plot of the exciton critical temperature for ${\mathrm{TiSe}}_2$ in the plane of the background dielectric constant ${\epsilon }_{\infty }$ and the normal gap $E_g$. The red line represents the region corresponding to $T_c=200$ K.

Figure 8

The exciton critical temperature for ${\mathrm{TiSe}}_2$ as a function of the normal gap $E_g$ from 3D gap equation.

Figure 9

Separate reflectance spectra of $1T\text{−}{\mathrm{TiSe}}_2$ at 300 K obtained from the optical spectroscopy [34] and the EELS study [35]. In the inset we show the combined reflectance.

Figure 10

The optical conductivity: the full conductivity (orange line) and a high-energy part above 6.5 eV (olive line).

Figure 11

The dielectric function ${\epsilon }_1\left(\omega \right)$ of $1T\text{−}{\mathrm{TiSe}}_2$ obtained from the KK analysis and the high-energy dielectric function, ${\epsilon }_1^H\left(\omega \right)$.

Figure 12

The conductivity ${\sigma }_1\left(\omega \right)$ and the high-energy part of the conductivity ${\sigma }_1^H\left(\omega \right)$ for temperatures 10 and 300 K. The ${\sigma }_1^H\left(\omega \right)$ for the two temperatures overlap exactly and, consequently, the background dielectric constant ${\epsilon }_{\infty }$ is temperature independent.

Figure 13

The dielectric function ${\epsilon }_1\left(\omega \right)$ and the high-energy part of the dielectric function ${\epsilon }_1^H\left(\omega \right)$ for the electric field parallel to the layers.

Figure 14

The full dielectric function and the high-energy part of the dielectric function for the electric field perpendicular to the layers.

Figure 15

(a) The $dI/dV$ of ${\mathrm{TiSe}}_2$ at 280 K (red) and 15 K (black) from Kolekar et al. [27]. (b) Normalized $dI/dV$.