Fermi-surface nesting and the origin of the charge-density wave in $\mathrm{Nb}{\mathrm{Se}}_2$

We use highly accurate density functional calculations to study the band structure and Fermi surfaces of . We calculate the real part of the noninteracting susceptibility, , which is the relevant quantity for a charge density wave (CDW) instability and the imaginary part, , which directly shows Fermi surface (FS) nesting. We show that there are very weak peaks in near the CDW wave vector, but that no such peaks are visible in , definitively eliminating FS nesting as a factor in CDW formation. Because the peak in is broad and shallow, it is unlikely to be the direct cause of the CDW instability. We briefly address the possibility that electron-electron interactions (local field effects) produce additional structure in the total (renormalized) susceptibility, and we discuss the role of electron-ion matrix elements.

Figure 1

(Color online) The crystal structure of $\mathrm{Nb}{\mathrm{Se}}_2$, showing the two distinct layers that make up the unit cell. Each Nb ion sits in a hexagonal mirror plane with three Se neighbors above and below, forming a network of distorted edge-sharing octahedra.

Figure 2

(Color online) The band structure of $\mathrm{Nb}{\mathrm{Se}}_2$. Top: A closeup of the bands around the Fermi energy, neglecting spin-orbit coupling. The highlighted bands are of Se $p_z$ character and cross the Nb bands without hybridization. Bottom: The full band structure, including the spin-orbit interaction which relieves the Nb band degeneracy in the $k_z=\pi ∕c$ plane. The highlighted bands are of combined $x^2-y^2∕xy$ character

Figure 3

(Color online) A schematic showing the origin of the splitting of the antibonding $p_z$ band and the Nb $d_{z^2-1}$ band at two different $k_z$ positions. The Nb ions are omitted from the structural graphic for clarity and the $d_{z^2-1}$ orbitals have been idealized as spheres. Left side: At the $\Gamma$-point, Nb $d_{z^2-1}$ orbitals are either fully bonding or fully antibonding, and the antibonding $p_z$ orbitals are antibonding even between cells (short arrows). Right side: At the $A$-point, there is no overall energy difference between bonding and antibonding Nb $d_{z^2-1}$ configurations and the antibonding $p_z$ orbitals are now bonding between cells. Plus signs indicate bonding and minus signs antibonding—the long arrows connect the topmost orbital to the one that would sit above it if the cells were repeated endlessly.

Figure 4

(Color online) The three Fermi surfaces of $\mathrm{Nb}{\mathrm{Se}}_2$ in an extended BZ scheme with the conventional BZ indicated as a hexagon in the center of each figure. (a) The small Se-derived (band 1) pancake surface around the $\Gamma$-point. (b) The bonding (band 2) Nb-derived Fermi surface with strong warping along $k_z$. (c) The nearly 2D anti-bonding (band 3) Nb-derived Fermi surface. The $\Gamma -K-M$ plane is at the top and bottom of each figure and the $A-L-H$ plane cuts through the center.

Figure 5

(Color online) The noninteracting susceptibility of $\mathrm{Nb}{\mathrm{Se}}_2$. (a) The imaginary part exhibits FS nesting driven peaks at $\mathbf{Q}=(\frac{1}{3},\frac{1}{3},0)$. The plane at the bottom is a guide for the eye and corresponds to the lowest value of $\mathrm{Im}{\chi }_0$. (b) The real part has very weak peaks at ${\mathbf{Q}}_{\mathit{CDW}}=(\frac{1}{3},0,0)$ that come from energy intervals away from $E_F$ (see text). The plane corresponds to the lowest value of $\mathrm{Re}{\chi }_0$, which is also the density of states at the Fermi level.

Figure 6

(Color online) $\mathrm{Re}{\chi }_0\left(\mathbf{q}\right)$ along the $\Gamma -M$ line of the Brillouin zone. The total is shown in shaded grey with the contributions from each individual band delineated by different lines. The height of band 1 has been multiplied by two to make it visible above the $x$-axis. The topmost line includes nondiagonal (interband) contributions.