Quasiparticle energy bands and Fermi surfaces of monolayer ${\mathrm{NbSe}}_2$

A quasiparticle band structure of a single layer $2H\text{−}{\mathrm{NbSe}}_2$ is reported by using first-principles $GW$ calculation. We show that a self-energy correction increases the width of a partially occupied band and alters its Fermi surface shape when comparing those using conventional mean-field calculation methods. Owing to a broken inversion symmetry in the trigonal prismatic single layer structure, the spin-orbit interaction is included and its impact on the Fermi surface and quasiparticle energy bands are discussed. We also calculate the doping dependent static susceptibilities from the band structures obtained by the mean-field calculation as well as $GW$ calculation with and without spin-orbit interactions. A complete tight-binding model is constructed within the three-band third nearest neighbor hoppings and is shown to reproduce our $GW$ quasiparticle energy bands and Fermi surface very well. Considering variations of the Fermi surface shapes depending on self-energy corrections and spin-orbit interactions, we discuss the formations of charge density wave (CDW) with different dielectric environments and their implications on recent controversial experimental results on CDW transition temperatures.

Figure 1

Crystal structure and Brillouin zone of a single layer of 2H-${\mathrm{NbSe}}_2$. (a) 2H structure of ${\mathrm{NbSe}}_2$. A transition metal atom (Nb, blue sphere) is surrounded by six chalcogens (Se, yellow sphere). Three of them lie on the upper plane, and the other three on the lower one. (b) Crystal structure from a top view. Primitive lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are denoted. (c) The first Brillouin zone and reciprocal lattice vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$.

Figure 2

Electronic band structures of monolayer ${\mathrm{NbSe}}_2$ along the IBZ boundary. Red solid line and blue dashed one represent DFT-GGA and $GW$ band structures, respectively. The Fermi energy is set to be zero. Black arrow indicates the saddle point of the partially occupied band.

Figure 3

Fermi surfaces of DFT-GGA (red solid line) (a) and $GW$ (blue dashed line) energy bands (b).

Figure 4

The real part of the noninteracting susceptibility ${\chi }_0^{\prime }\left(q\right)$ of (a) DFT-GGA and (b) $GW$ bands within the constant matrix approximation (in arbitrary units). The susceptibility is calculated at $k_{B}T=10\mathrm{meV}$.

Figure 5

The real part of the noninteracting susceptibility ${\chi }_0\left(q\right)$ of (a) DFT-GGA and (b) $GW$ bands along the IBZ boundary ($\mathrm{\Gamma }MK$) for several temperatures $[k_{B}T=10$ (purple), 20 (blue), 30 (yellow), 40 (green), and 50 (black) meV]. To clarify, the curves are shifted vertically by proper amounts.

Figure 6

Fermi surfaces (blue solid lines for the hole doped case and red for electron) of the single layer where the Fermi energy is rigidly shifted by $\delta {\epsilon }_F$ due to doping from the charge neutral one (black): (a) $\delta {\epsilon }_F=-150\mathrm{meV}$, (b) $\delta {\epsilon }_F=-77\mathrm{meV}$, (c) $\delta {\epsilon }_F=0\mathrm{meV}$, (d) $\delta {\epsilon }_F=80\mathrm{meV}$, and (e) $\delta {\epsilon }_F=150\mathrm{meV}$.

Figure 7

${\chi }_0^{\prime }\left(\mathbf{q}\right)$ of $p$-doped cases for $GW$ bands: (a) $\delta {\epsilon }_F=-77\mathrm{meV}$ and (b) $\delta {\epsilon }_F=-150\mathrm{meV}$. ${\chi }_0^{\prime }\left(\mathbf{q}\right)$ of $n$-doped cases for $GW$ bands: (c) $\delta {\epsilon }_F=80\mathrm{meV}$ and (d) $\delta {\epsilon }_F=150\mathrm{meV}$.

Figure 8

Partially occupied bands splitted by SOC along the IBZ boundary: (a) DFT-GGA and (b) $GW$ bands. Red dashed lines and blue solid ones indicate electronic bands without and with SOC, respectively.

Figure 9

Fermi surfaces of electronic bands corrected by SOC: (a) DFT-GGA and (b) $GW$ bands. Red dashed lines and blue solid ones indicate Fermi surfaces without and with SOC, respectively.

Figure 10

The real part of the noninteracting susceptibility ${\chi }_0^{\prime }\left(q\right)$ of SOC-corrected electronic bands from (a) DFT-GGA and (b) $GW$ methods.

Figure 11

(a) $GW$ bands for a single layer ${\mathrm{NbSe}}_2$ with and without semicore levels of $4s^{2}4p^6$ of Nb atom, which are denoted by blue solid lines and red dashed ones. (b) Fermi surfaces of $GW$ bands with semicore states (blue solid lines), $GW$ without semicore states (red dashed lines), and DFT-GGA bands (black dotted lines).

Figure 12

(a) Electronic bands from the DFT-GGA calculation (blue dashed line) and ones from the three-band TNN TB model fit to the DFT-GGA bands (red solid line). (b) Fermi surfaces of electronic bands from the DFT-GGA calculation (blue dashed line) and the TB model (red solid line). (c) Electronic bands from the $GW$ approximation (blue dashed line) and ones from the three-band TNN TB model fit to the $GW$ bands (red solid line). (d) Fermi surfaces of electronic bands from the $GW$ calculation (blue dashed line) and the TB model (red solid line).

Figure 13

Orbital projected band structures from the three-band TNN TB model of the $GW$ bands: contributions from (a) the $d_{z^2}$ orbital (blue circle) and (b) $d_{xy}$ and $d_{x^2-y^2}$ orbitals (red circle). The size of open circle is proportional to contribution of the corresponding orbital. The Fermi energy (blue dashed line) is set to be zero.

Figure 14

(a) Energy bands of the bulk $2H\text{−}{\mathrm{NbSe}}_2$ along symmetric lines of the IBZ. Red dashed lines and blue solid lines correspond to DFT-GGA and $GW$ electronic bands, respectively. (b) and (c) are Fermi surfaces of DFT-GGA bands on $\mathrm{\Gamma }MK$ and $ALH$ planes of the IBZ, respectively. Similarly (d) and (e) show Fermi surfaces of $GW$ bands on $\mathrm{\Gamma }MK$ and $ALH$ planes, respectively.