Role of band structure and local-field effects in the low-energy collective electronic excitation spectra of 2$H$-NbSe${}_2$

We present a study of the electron dynamics in the layered compound 2$H$-NbSe${}_2$. First-principles calculations are used to obtain the band structure employed in the evaluation of the loss function with inclusion of local-field (LF) effects. Two different symmetry directions [(100) and (010)] were explored in the hexagonal basal plane. In both cases, a low-energy charge-carrier plasmon (CCP) at $\sim$1 eV presenting a negative dispersion over a wide momentum transfer range is found, in agreement with recent experimental results [Wezel et al., Phys. Rev. Lett. 107, 176404 (2011)]. On the contrary, in the (001) perpendicular direction, the CCP has negative dispersion at small momenta only, presenting strong positive dispersion at larger momenta. Our calculations reveal that this behavior can be explained without invoking many-body effects, as long as band structure effects are properly included in the evaluation of the excitation spectra. In addition to this CCP mode, we find another one with an arclike oscillating dispersion along the perpendicular direction, as well as the appearance of a CCP replica at high momenta due to LF effects.

Figure 1

(a) Projection onto a hexagonal basal plane of atomic positions of the 2$H$-NbSe${}_2$ unit cell. The Nb ion positions at $z=c/4$ and $z=-c/4$ are shown by the large gray filled circle. The Se ions are shown by smaller filled circles. The planes containing the Se ions located at $z=0.1183c$ and $z=-0.3817c$ ($z=-0.1183c$ and $z=0.3817c$) are shown by red (blue) color circles. (b) The first Brillouin zone (1BZ) with main symmetry directions for the in-plane momentum transfer. The (001) axis is directed in the perpendicular direction. The $\Gamma$KM (AHL) plane is located at $k_z=0$ ($k_z=\pi /c$). In this paper we use the convention wherein the (100), (010), and (001) momentum transfer axes are directed along the $\Gamma$K, $\Gamma$M, and $\Gamma$A directions of the 1BZ, respectively. (c) The calculated band structure along some symmetry directions of the 1BZ around the Fermi level which set at the zero energy. The three energy bands in which the intraband transitions are available are shown by thick red lines and labeled as 1, 2, and 3.

Figure 2

2D plots of calculated energy-loss function of 2$H$-NbSe${}_2$ versus momentum transfer $Q$ along the (100) symmetry direction and energy $\omega$, calculated with (a) inclusion and (b) without inclusion of local-field effects. The data are obtained with the RPA kernel. The dominating peak corresponding to the charge-carrier plasmon (CCP) is highlighted by the corresponding symbol. The experimentally measured (Ref. 14) energy positions of peaks corresponding to the CCP are shown by filled circles. The imaginary part of $\epsilon (Q,\omega )$ evaluated with inclusion of both intraband and interband transitions is presented in (c) on a logarithmic scale. Arrow in (a) shows a satellite plasmon peak having its origin in the enhanced interband transitions peak in Im$\left[\epsilon \right]$ marked by an arrow in (c).

Figure 3

Similar to Fig. 2 for the (010) symmetry direction. Here, $a^{*}=a\sqrt{3}$.

Figure 4

Similar to Fig. 2 for the (001) symmetry direction. In (a) the acoustic plasmon arclike dispersion is highlighted by thick dotted lines. Thin short-dashed lines delimit regions for intraband transitions within the Nb-originated bands labeled as 2 and 3 in Fig. 1c. In (c) peaks in Im$\left[\epsilon \right]$ due to intraband transitions within the Se-originated band 1 and the Nb-derived bands 2 and 3 are marked as IT1 and IT2, respectively.

Figure 5

Imaginary (long-dashed line) and real (dotted line) parts of the dielectric function $\epsilon$ and corresponding energy-loss function Im$\left[{\epsilon }^{-1}\right]$, with inclusion (solid line) and without (dashed line) inclusion of the local-field effects, calculated for momentum transfer $Q=0.152{\AA }^{-1}$ along the (100) symmetry direction. Data obtained with RPA kernel are presented. The ALDA curves are undistinguished from the RPA ones on the presented scale.

Figure 6

2D plots of calculated energy-loss function of 2$H$-NbSe${}_2$ versus momentum transfer $Q$ along the (100) symmetry direction and energy $\omega$, calculated with (a) inclusion and (b) without inclusion of local-field effects. The intraband transitions are only included in the evaluation of ${\chi }^0$. The data are obtained with the RPA kernel. A dominating peak corresponding to the charge-carrier plasmon (CCP) is highlighted by the corresponding symbol. Imaginary part of $\epsilon (Q,\omega )$ is presented in (c) on a logarithmic scale.

Figure 7

Similar to Fig. 6 for (010) symmetry direction. Here, $a^{*}=a\sqrt{3}$.

Figure 8

Similar to Fig. 6 for (001) symmetry direction.

Figure 9

Similar to Fig. 5 calculated for momentum transfer $Q=0.189{\AA }^{-1}$ along the (001) symmetry direction. Circles highlight the zero crossing in the real part of $\epsilon$. Only the zero crossings denoted by filled dots produce a peak in the energy-loss function, whereas the crossings shown by open circles do not produce any effect due to peaks in the imaginary part of $\epsilon$ at fairly the same energies.

Figure 10

Similar to Fig. 5 calculated for momentum transfer $Q=0.881{\AA }^{-1}$ along the (001) symmetry direction. Thin (thick) lines stand for the data obtained in RPA (ALDA) calculations.

Figure 11

Similar to Fig. 5 calculated for momentum transfer $Q=2.02{\AA }^{-1}$ along the (010) symmetry direction. Thin (thick) lines stand for the data obtained in RPA (ALDA) calculations.