Spontaneous exciton condensation in $1T{\text{-TiSe}}_2$: BCS-like approach

Recently we found strong evidence in favor of a BCS-like condensation of excitons in $1T{\text{-TiSe}}_2$ [Cercellier et al., Phys. Rev. Lett. 99, 146403 (2007)]. Theoretical photoemission intensity maps have been generated by the spectral function calculated within the exciton condensate phase model and set against experimental angle-resolved photoemission spectroscopy data. The scope of this paper is to present the detailed calculations in the framework of this model. They represent an extension of the original excitonic insulator phase model of Jérome et al. [Phys. Rev. 158, 462 (1967)] to three dimensional and anisotropic band dispersions. A detailed analysis of its properties and comparison with experiment is presented. Finally, the disagreement with density-functional theory is discussed.

Figure 1

Schematic picture of the $1T{\text{-TiSe}}_2$ bands considered in this model (near the Fermi energy $E_F$). (a) Top view of the BZ (perpendicular to $k_z$). The Fermi surface has a hole pocket at $\Gamma$ and three symmetry equivalent electron pockets at $L$, separated from $\Gamma$ by the spanning vectors ${\vec{w}}_i,i=1,2,3$. The side view of the BZ helps to situate the high-symmetry points. (b) Schematic cut along the $\Gamma L$ direction, showing the dispersions of the valence band (at $\Gamma$) and one conduction band (at $L$).

Figure 2

(Color online) Schematic picture of the band positions in the model (near $E_F$) in the normal phase and in the CDW phase. Wave vectors are expressed in a multiple of $\Gamma M$. (a) In the normal phase $\left(\Delta =0\text{meV}\right)$, the Fermi surface composed of the valence band at $\Gamma$ (in red) and three symmetry equivalent conduction bands at $L$ (in blue). (b) In the CDW phase $\left(\Delta \ne 0\text{meV}\right)$, $\Gamma$ becomes equivalent to $L$. The electron pockets at $L$, backfolded to $\Gamma$, produce “flowerlike” Fermi surfaces at each newly equivalent high-symmetry point. (c) Dispersions calculated parallel to $\Gamma M$ [see Fig. 1a] around $\Gamma$ and parallel to $A{L}_1$ around the three $L$ points in the normal phase plotted on the same graph (the minima of the different conduction bands $c_1,c_2,c_3$ have been displaced from the $L$ points to $\Gamma$ on the graph). (d) and (e) Dispersions around $\Gamma$ and along $\Gamma M$ in the CDW phase for $\Delta =20\text{meV}$, respectively, $\Delta =100$. In the CDW phase, $\Gamma$ and $L$ become equivalent concerning the dispersions.

Figure 3

(Color online) Extrema of the renormalized bands as a function of the order parameter $\Delta$ (left). Position of these extrema on the band dispersions (right).

Figure 4

Band dispersions with their corresponding spectral weight at $\Gamma$ and $L$, along $\Gamma M$ and $AL$ directions, respectively. Graph (a) describes the normal phase $\left(\Delta =0\text{meV}\right)$, (b) the CDW phase with moderate excitonic effects $\left(\Delta =20\text{meV}\right)$, and (c) the CDW phase with strong excitonic effects $\left(\Delta =100\text{meV}\right)$. The dashed lines indicate a band $\left(c_1\right)$ having a small nonzero SW (see text).

Figure 5

(Color online) Spectral weights of the bands at $\Gamma$ and $L$, along $\Gamma M$ and $AL$ directions, respectively, for $\Delta$ values of 0, 20, and 100 meV. Graphs (a) and (b) describe the valence band $\left(v_1\right)$ at $\Gamma$ and its backfolded version at $L$, respectively. Graphs (c) and (d) describe the conduction band $c_3$ at $\Gamma$ (where it follows the top of the original valence band) and $L$, respectively. Graphs (e) and (f) describe the conduction band $c_1$ at $\Gamma$ and $L$, respectively (where it is the original conduction band).

Figure 6

(Color online) Comparisons between theoretical and experimental $\left(h\nu =31\text{eV}\right)$ electronic structures at $\Gamma$ and $\overline{\Gamma }$, respectively (see text for explanations of this notation). (a) The theoretical bands have been calculated for $\Delta =25\text{meV}$ and the experimental intensity maps are taken at $T=250\text{K}$. The continuous black lines highlight the $\text{Se}4p$-derived bands not considered in the model, while the dashed white line indicates the valence band corresponding to $v_1$. (b) The theoretical bands have been calculated for $\Delta =75\text{meV}$ and the experimental intensity maps are taken at $T=65\text{K}$. The dashed black lines indicate the backfolded conduction band $c_1$ which carries a small nonzero SW [Fig. 5e].

Figure 7

(Color online) Comparisons between theoretical and experimental $\left(h\nu =31\text{eV}\right)$ electronic structures at the Brillouin-zone boundary (see text). (a) The theoretical bands have been calculated for $\Delta =25\text{meV}$ and the experimental intensity maps are taken at $T=250\text{K}$. The dashed white line indicates the conduction band corresponding to $c_1$. (b) The theoretical bands have been calculated for $\Delta =75\text{meV}$ and the experimental intensity maps are taken at $T=65\text{K}$.

Figure 8

Band dispersions of the ideal (nondoped) system (see text) with their corresponding spectral weight at $\Gamma$ and $L$, along $\Gamma M$ and $AL$ directions, respectively. An order parameter of $\Delta =20\text{meV}$ has been used, describing a CDW phase with moderate excitonic effects.