Phonon-Assisted Magnetic Mott-Insulating State in the Charge Density Wave Phase of Single-Layer $1T\text{−}{\mathrm{NbSe}}_2$

We study the structural, electronic, and vibrational properties of single-layer $1T\text{−}{\mathrm{NbSe}}_2$ from first principles. Within the generalized gradient approximation, the $1T$ polytype is highly unstable with respect to the $2H$. The density functional theory with the Hubbard $U$ method improves the stability of the $1T$ phase, explaining its detection in experiments. A charge density wave occurs with a $\sqrt{13}\times \sqrt{13}R30°$ periodicity, in agreement with STM data. At $U=0$, the David-star reconstruction displays a flatband below the Fermi level with a marked $d_{z^2-r^2}$ orbital character of the central Nb. The Hubbard interaction induces a magnetic Mott-insulating state. Magnetism distorts the lattice around the central Nb atom in the star, reduces the hybridization between the central Nb $d_{z^2-r^2}$ orbital and the neighboring Se $p$ states, and lifts in energy the empty $d_{z^2-r^2}$ flatband becoming nonbonding. This cooperative Jahn-Teller and correlation effect amplifies the Mott gap. Our results are relevant for the broad class of correlated insulator in the presence of a strong Jahn-Teller effect.

Figure 1

(a) Crystal structure of an undistorted $1T\text{−}{\mathrm{NbSe}}_2$ single layer (Nb atoms in gray, Se atoms in yellow). In the bottom image, the octahedral coordination around the Nb atom is shown. Electronic structure (b) and phonon dispersion (c) of single-layer $1T\text{−}{\mathrm{NbSe}}_2$ in the high-symmetry phase. In (b) SOC (no SOC) means with (without) inclusion of spin-orbit coupling. In the nonrelativistic case, the size of the blue dots is proportional to the $3d_{z^2-r^2}$ orbital component (at $\mathbf{\Gamma }$, the larger dot corresponds to a 53% $d_{z^2-r^2}$ component).

Figure 2

(Left) Optimized crystal structure for the low-temperature phase of $1T\text{−}{\mathrm{NbSe}}_2$. There are three inequivalent Nb sites: central Nb (red), hexagonal site belonging to the $\sqrt{7}\times \sqrt{7}$ cluster (blue), and peripheral Nb sites (gray). The Se atoms are shown in yellow. The red line is a guide to the eye to recognize the $\sqrt{13}\times \sqrt{13}$ cluster (David star). The inclusion of the Hubbard term leads to a ferromagnetic state with an important local distortion around the central Nb atom in the star. However, the difference between the ferromagnetic and nonmagnetic structures are not visible on this scale. The magnetic state has a total magnetization of $+1{\mu }_B$. The magnetic moments on Nb atoms are $0.58{\mu }_B$ central (red) Nb, 0.0417 (blue) Nb, $-0.0256$ peripheral (gray) Nb. The moduli of the magnetic moments on Se atoms are smaller than $0.08{\mu }_b$. (Center and left) Magnetic-induced distortion (side and top views) in the central $\sqrt{7}\times \sqrt{7}$ cluster. The central atom is unshifted. The six blue Nb atoms interacting ferromagnetically with the central one moves apart in plane. The out-of-plane Se approaches the Nb basal plane. In the top view (right) the bottom Se atoms are shown in a darker yellow color.

Figure 3

(Left) Electronic structure in the charge density wave phase of $1T\text{−}{\mathrm{NbSe}}_2$ within the nonmagnetic GGA. The size of the red dots is proportional to the Nb central-atom $d_{z^2-r^2}$ orbital character (the red dot at zone center and $-0.05\mathrm{eV}$ corresponds to 15% orbital character). (Center) Same as the left but in the $\mathrm{GGA}+U$ approximation on top of the nonmagnetic geometry. Continuous (dashed) lines are majority (minority) spin bands. The size of the full (open) dots is proportional to the Nb central-atom $d_{z^2-r^2}$ up spin (down spin) orbital character. (Right) As in the center, but on top of the $\mathrm{GGA}+U$ geometry. The Hubbard-induced distortion raises in energy the minority spin band and increases the Mott gap.