Anomalous Dirac Plasmons in 1D Topological Electrides

The plasmon opens up the possibility to efficiently couple light and matter at subwavelength scales. In general, the plasmon frequency, intensity, and damping are dependent on the carrier density. These dependencies, however, are disadvantageous for stable functionalities of plasmons and render fundamentally a weak intensity at low frequency, especially for the Dirac plasmon (DP) widely studied in graphene. Here we demonstrate a new type of DP, emerging from a Dirac nodal-surface state, which can simultaneously exhibit a density-independent frequency, intensity, and damping. Remarkably, we predict the realization of anomalous DP (ADP) in 1D topological electrides, such as ${\mathrm{Ba}}_3{\mathrm{CrN}}_3$ and ${\mathrm{Sr}}_3{\mathrm{CrN}}_3$, by first-principles calculations. The ADPs in both systems have a density-independent frequency and high intensity, and their frequency can be tuned from terahertz to midinfrared by changing the excitation direction. Furthermore, the intrinsic weak electron-phonon coupling of anionic electrons in electrides affords an added advantage of low-phonon-assisted damping and hence a long lifetime of the ADPs. Our Letter paves the way to developing novel plasmonic and optoelectronic devices by combining topological physics with electride materials.

Figure 1

(a) Density dependences of plasmon frequency and intensity. (Green curve) The usual plasmon following a $\omega \sim n^{\alpha }$ scaling ($\alpha =1/2,1/3,1/4$ for parabolic metals, DNP, and DNL semimetals, respectively). (Red curve) Our proposed ADP following the $\omega \sim n^0$ scaling. The brightness of the color indicates the intensity of the plasmon [15]. (b) (Left) Schematic plot for 1D electride, where the orange channels denote the anionic electrons. (Inset) A band crossing of a single anionic electron chain along the $k_z$ direction. (Right) The crossing points form a 2D degenerate DNS. (c) Schematic illustration of the phonon-assisted damping for the Dirac plasmon. The plasmon may enter the intraband Landau region and decay into electron-hole pairs via phonon scattering or by emitting a phonon. This damping pathway is naturally suppressed in electrides.

Figure 2

(a) Top (upper) and side (lower) view of the crystal structure of ${\mathrm{Ba}}_3{\mathrm{CrN}}_3$. (b) The first BZ of ${\mathrm{Ba}}_3{\mathrm{CrN}}_3$. The high-symmetry and some arbitrary points (dots) are labeled. (c) Band structure with atomic orbital projections. The Fermi level is set to zero. The partial charge density distribution for the energy bands around $E_F$ ($-0.3–0.3\mathrm{eV}$) are depicted by the red regions in (a). (d) Magnified band structure along an arbitrary line $P_1–P–P_2$ perpendicular to the $k_z=\pi /c$ plane at an arbitrary $P$ point located in the plane, as indicated in (b). The nodal surface in momentum space is plotted by the orange plane ($k_z=\pi /c$) in (b).

Figure 3

(a) Real and imaginary parts of dielectric function and EELS of ${\mathrm{Ba}}_3{\mathrm{CrN}}_3$ as functions of frequency with $q=0.001{\AA }^{-1}$ along the $z$ direction. (b) EELS as a function of frequency $\hslash \omega$ and wave vector $q_z$. The white dashed line denotes the upper edge $\hslash v_{F}q$ of the intraband particle-hole continuum. (c) Plasmon frequency as a function of $\mathrm{\Delta }n$ and excitation direction ($\theta$). The color from red to yellow indicates the frequency from terahertz to mid-IR, and the brightness of color indicates almost a constant high plasmon intensity.

Figure 4

Average $e$-ph coupling matrix element $|g_{mn,\nu }(\mathit{k},\mathit{q})|$ of all 42 phonon modes for the DNS band of ${\mathrm{Ba}}_3{\mathrm{CrN}}_3$ at the $A$ point of BZ versus phonon wave vector $\mathit{q}$ in the (a) $q_z=0$ and (b) $q_x=0$ plane. (c) Electron linewidth for the DNS band of anionic electrons (red line) and the higher band of atomic bound electrons (black line) along high-symmetry $\mathit{k}$ points. (d) Scattering rate of electron versus energy at 0 K (blue dots) and 300 K (red dots), respectively. The Fermi level is set to zero. The three black triangles in (d) and enlarged view in the inset are the calculated data of phonon-assisted plasmon damping rate at 300 K, indicating that plasmon damping rates are lower than scattering rates (see Refs. [27, 57, 58, 59, 60] for details).