Effect of Charge Order on the Plasmon Dispersion in Transition-Metal Dichalcogenides

We investigate the dispersion of the charge carrier plasmon in the three prototypical charge-density wave bearing transition-metal dichalcogenides $2H\mathrm{\text{−}}{\mathrm{TaSe}}_2$, $2H\mathrm{\text{−}}{\mathrm{TaS}}_2$, and $2H\mathrm{\text{−}}{\mathrm{NbSe}}_2$ employing electron energy-loss spectroscopy. For all three compounds the plasmon dispersion is found to be negative for small momentum transfers. This is in contrast with the generic behavior observed in simple metals as well as the related system $2H\mathrm{\text{−}}{\mathrm{NbS}}_2$, which does not exhibit charge order. We present a semiclassical Ginzburg-Landau model which accounts for these observations, and argue that the vicinity to a charge ordered state is thus reflected in the properties of the collective excitations.

Figure 1

The EELS intensity for the three investigated compounds measured at room temperature. The spectra have been normalized on the high-energy side and the data for $2H\mathrm{\text{−}}{\mathrm{TaSe}}_2$ are partially reproduced from Ref. 17.

Figure 2

Left: The plasmon dispersions for $2H\mathrm{\text{−}}{\mathrm{TaS}}_2$, $2H\mathrm{\text{−}}{\mathrm{TaSe}}_2$, and $2H\mathrm{\text{−}}{\mathrm{NbSe}}_2$ as extracted from the peak positions in Fig. 1 and for $2H\mathrm{\text{−}}{\mathrm{NbS}}_2$ reproduced from Ref. 31. The energies have been scaled with respect to the plasma energy for each material, and the momentum transfers with respect to the CDW ordering vector. Right: Temperature dependence of the plasmon bandwidth for $2H\mathrm{\text{−}}{\mathrm{TaSe}}_2$, defined as the difference of the plasmon energy between $q=0.1{\AA }^{-1}$ and $q=0.5{\AA }^{-1}$. The vertical line indicates the onset of the incommensurate CDW.

Figure 3

The plasmon dispersions of Eq. (4). Depending on the values of the model parameters, the dispersion can be either uniformly positive or have a negative dispersion at low $k$ followed by a minimum. The topmost, dashed line shows the shape of a bare dispersion with $a=b=0$ and $\alpha =0.75{\omega }_p^2/Q^2$, for an arbitrary choice of $Q$. The second line is the dispersion for $a=-\alpha n^{\prime }m/2$, the value at which the second minimum first appears. Here we used $c=b{Q}^4=0.3n^{\prime }m{\omega }_p^2$. The line on the bottom shows the situation with a well-developed minimum for the same value of $c$ and $a=-\alpha n^{\prime }m/2-\sqrt{bc}$.