Tilt-induced kink in the plasmon dispersion of two-dimensional Dirac electrons

The list of two-dimensional Dirac systems with a tilt in their Dirac cone spectrum is expanding, and now, in addition to the organic system $\alpha {(\mathrm{BEDT}-\mathrm{TTF})}_2{\mathrm{I}}_3$, it includes the two-dimensional $8Pmmn$-borophene sheet, which allows for controlled doping by the gate voltage. We analytically calculate the polarization function of tilted Dirac cone for an arbitrary tilt parameter, $0\le \eta <1$, and arbitrary doping. This enables us to find two interesting plasmonic effects solely caused by the tilt. (i) In addition to the standard plasmon oscillations, a strong enough tilt induces an additional linearly dispersing overdamped branch of plasmons, which is strongly Landau damped due to overlap with a large density of intraband free particle-hole (PH) excitations. (ii) There appears a kink in the plasmon dispersion for any nonzero tilt parameter. The kink appears when the plasmon branch enters the interband continuum of PH excitations. This kink becomes most manifest for wave vectors perpendicular to the tilt direction and fades away by approaching the tilt direction. Experimental measurements of the wave vector and energy of the plasmonic kink, when combined with our analytic formula for the kink energy scale, allow for a direct experimental estimation of the tilt parameter. Furthermore, we find that, for a fixed chemical potential $\mu$, the screening is significantly enhanced by bringing the tilt $\eta$ close to 1.

Figure 1

Different regions in the space of $q/k_F$ and $\hslash \omega /\mu$. Various regions determine the sign structure coming from step functions in the tilted Dirac cone. This figure is produced for the tilt parameter $\eta =0.45$ and $\varphi =\pi /3$, which will be extensively used in this paper. In Ref. [34], a similar figure is produced for $\eta =0.8$. In the limit $\eta \to 0$, this figure becomes identical to Fig. 6 of Ref. [33]. Part of the dashed border that separates region 1B and 2B is where the plasmon kink develops (see Sec. 4 of the main text).

Figure 2

Comparison of real and imaginary parts of the polarization function for a doped tilted Dirac cone with the result in Ref. [34]. The vertical axis is in units of $\mu /{\hslash }^2{v}_x{v}_y$ and $v_{x0}=0.8$ and $v_{y0}=0$, i.e., $\eta =0.8$. The momentum $q$ is in the $y$ direction ($\varphi =\pi /2$) and equals to 4. The blue (solid) and red (dot-dashed) curves correspond to our calculation and the result in Ref. [34], respectively.

Figure 3

Real part of the polarization function for a doped tilted Dirac cone, for different values of $\eta$ as indicated in the legend. The direction of $\mathit{q}$ is fixed by $\varphi =\pi /2$. The vertical axis is in the unit $\mu /{\hslash }^2{v}_x{v}_y$ and the horizontal axis is a dimensionless quantity $\hslash \omega /\mu$. The magnitude of $q/k_F$ in the various panels is 0.25 in (a), 0.5 in (b), 0.75 in (c), and 2.0 in (d). The solid line is the imaginary part of $\chi$ for $\eta =0.9$. For explanation see the text.

Figure 4

The overdamped plasmon mode arising from tilt for the direction $\varphi =\pi /2$. Left (right) panel corresponds to tilt parameter $\eta =0.75$ ($\eta =0.9$). The shaded region represents damping. The amount of damping in the right panel is decreased by a factor of 10 to fit in the panel.

Figure 5

Real part of $\chi$ in units of $\mu /{\hslash }^2{v}_x{v}_y$ as a function of dimensionless $\hslash \omega /\mu$ for fixed $q/k_F=0.5$ and various values of angle $\varphi$ indicated in the legend. The left (right) panel corresponds to $\eta =0.45$ ($\eta =0.9$).

Figure 6

Constant energy cuts of the plasmon dispersion for a moderate tilt parameter $\eta =0.45$. Curves are generated for $v_x=v_y=c/1000$.

Figure 7

Dependence of plasmon dispersion on the tilt parameter $\eta$ and angle $\varphi$. The horizontal axis is $q/k_F$ and the vertical axis is $\hslash \omega /\mu$. The tilt parameters $\eta =0,0.3,0.45,0.6,0.9$ are encoded as solid (red), dashed (green), dotted (blue), dot-dashed (purple), and the leftmost solid (black) curves, respectively. (a)–(f) correspond to directions $\varphi =\pi n/5$, with $n=0,...,5$.

Figure 8

Angular dependence of the plasmonic kink. The figure is produced for the moderate tilt parameter $\eta =0.45$, pertinent to realistic materials [18].

Figure 9

The position of the kink for the tilt parameter $\eta =0.45$. The solid (blue) line is the plasmon dispersion, and the dashed (red) curve is the ${\omega }_A\left(q\right)$ given by Eq. (23). (a)–(d) correspond to $\varphi =n\pi /5$ with $n=1,2,3,4$, respectively. Note that the width of the plasmon dispersion in region 2B is exaggerated by a factor of 50 to emphasize very small damping.

Figure 10

Imaginary part of the loss function $\mathrm{Im}|1/\varepsilon (\mathit{q},\omega \left)\right|$ and the dispersion of the plasmons for different tilt parameters and fixed momentum direction $\varphi =\pi /2$. The solid line is the plasmon dispersion, and the dashed curve denotes the borders of the PHC. The lower bound of the interband PHC is the ${\omega }_{\mathrm{kink}}$, which is given by Eq. (23). The tilt parameter in (a)–(d) is $\eta =0.3,0.45,0.6,0.9$, respectively.