Three-dimensional Fermi surface of $2H-\mathrm{NbS}{\mathrm{e}}_2$: Implications for the mechanism of charge density waves

We investigate the three-dimensional electronic structure of the seminal charge-density-wave (CDW) material $2H-\mathrm{NbS}{\mathrm{e}}_2$ by soft x-ray angle-resolved photoelectron spectroscopy and density functional theory. Our results reveal the pronounced 3D character of the electronic structure formed in the quasi-two-dimensional layered crystal structure. In particular, we find a strong dispersion along $k_z$ excluding a nesting-driven CDW formation based on experimental data. The 3D-like band structure of $2H-\mathrm{NbS}{\mathrm{e}}_2$ has strong implications for the intriguing phase competition of CDW order with superconductivity.

Figure 1

(a) BZ of $2H-\mathrm{NbS}{\mathrm{e}}_2$. Capital letters denote high-symmetry points. (b), (c) Comparison of calculated band structure (dashed lines) and SX-ARPES intensities measured at $hv=632$ eV along the $\mathrm{\Gamma }-M$ direction distinguished for (b) symmetric and (c) antisymmetric states employing $p$- and $s$-polarized light (see text). Two Nb $d_{3z^2-r^2}$ bands and one Se $p_z$ band making up the Fermi surface are shown as solid lines. (d) The same comparison along the $\mathrm{\Gamma }-K$ direction for the symmetric states employing $p$-polarized light (see text). (e) Same direction and $p$ polarization as in (b) but measured with higher $hv=732$ eV, corresponding to ${\mathrm{\Gamma }}_{28}$, and (f) its zoom-in around the $\mathrm{\Gamma }$ point. The Se $p_z$ bands clearly cross $E_F$ to form the 3D “pancake” FS pocket.

Figure 2

Comparison of calculated Fermi surface (lines) and SX-ARPES intensities observed in the $\mathrm{\Gamma }KM$ plane of the BZ with (a) $k_z\approx 13{\AA }^{-1}$ achieved at $hv=634$ eV and (b) $13.5{\AA }^{-1}$ at 674 eV, and (c) intensities in the ALH plane of the BZ with $k_z\approx 12.75{\AA }^{-1}$ at $hv=609\mathrm{eV}$. High-symmetry points are indicated by (gray) capital letters.

Figure 3

Fermi surface in $k_x-k_z$ planes for three different values of $k_y$ indicated by the dashed lines in (a) plot of intensities in the $k_x-k_y$ plane ($k_z\approx 13{\AA }^{-1}$) [same data as in Fig. 2]. (b)–(e) Comparison between calculated band structure (lines) and observed SX-ARPES intensities for $k_x-k_z$ planes including (b) the $\mathrm{\Gamma }\left(A\right)$ point, (d) the $K\left(H\right)$ point, and (e) for the direction indicated in (a) which was proposed to be relevant for FS nesting in Ref. [14]. In (e), the ordering wave vector ${\mathbf{q}}_{\mathrm{CDW}}$ (gray arrow) connects the FS dispersions only in a narrow $k_z$ range and can therefore not be related with any FS nesting. (c) The band dispersion along the $\mathrm{\Gamma }-A$ line [corresponding to $k_x=k_y=0$ in panel (b)] that shows the fourfold reduced $k_z$ periodicity.

Figure 4

(a) Calculated 3D FS. Capital letters denote high-symmetry points of the Brillouin zone. White arrows connect parts of the inner $K$ pocket, where nesting was proposed [14] and CDW hotspots were reported [15]. (b) Calculated electronic joint density of states (eJDOS) corresponding to the FS shown in (a).

Figure 5

Illustration of the model of Doran [19] considering the FS body corresponding to the inner $K$ pocket. $\mathbf{q}$ denotes the wave vector by which the FS body on the right is shifted with regard to the one on the left. ${\mathbf{v}}_{\mathbf{k}}$ and ${{\mathbf{v}}_{\mathbf{k}}}_{+\mathbf{q}}$ and the corresponding arrows indicate the Fermi velocities at one intersection point of the original and shifted FS bodies, respectively, and $\mathrm{\Delta }\mathbf{v}={\mathbf{v}}_{\mathbf{k}}-{{\mathbf{v}}_{\mathbf{k}}}_{+\mathbf{q}}$.