Collective charge excitations of the two-dimensional electride ${\mathrm{Ca}}_2\mathrm{N}$

${\mathrm{Ca}}_2\mathrm{N}$ is a layered material that has been recently identified as a two-dimensional (2D) electride, an unusual ionic compound in which electrons serve as anions. The electronic properties of 2D electrides attract considerable interest as the anionic electrons, which form a 2D layer sandwiched between atomic planes, are highly mobile as they are not attached to any ion. Here, on the basis of first-principles time-dependent density-functional theory calculations, we investigate the collective excitations of the electrons—i.e., the plasmons—in ${\mathrm{Ca}}_2\mathrm{N}$ as a function of wave vector $\mathbf{q}$. Our calculations reveal an intrinsic negative in-plane dispersion of the anionic plasmon, in striking contrast with the homogeneous electron gas. Moreover, for wave vectors $\mathbf{q}$ normal to the planes, we find a long-lived plasmon that continues to exist well beyond the first Brillouin zone. This is a mark of the electronic inhomogeneities in the charge response that ${\mathrm{Ca}}_2\mathrm{N}$ shares with other layered materials like transition-metal dichalcogenides and ${\mathrm{MgB}}_2$. Finally, we compare the plasmon properties of ${\mathrm{Ca}}_2\mathrm{N}$ in its bulk and monolayer forms, which shows the effect of the different electronic structures and dimensionalities.

Figure 1

(a) Crystal structure of ${\mathrm{Ca}}_2\mathrm{N}$ in a cut normal to the atomic planes. In the ball-and-stick representation grey spheres are Ca atoms and red spheres are N atoms. Depicted in light blue is the excess electronic charge associated to a wave function at the Fermi level. (b) Band structure in the rhombohedral Brillouin zone. For the high-symmetry $\mathbf{k}$ points we follow the notation of Ref. [18]. The zero of the energy is the Fermi level. The anionic band is marked in red.

Figure 2

Loss function of ${\mathrm{Ca}}_2\mathrm{N}$ as a function of in-plane momentum transfer $\mathbf{q}$.

Figure 3

(a) Intraband loss function of ${\mathrm{Ca}}_2\mathrm{N}$ as a function of in-plane momentum transfer $\mathbf{q}$ showing to the intrinsic anionic plasmon dispersion. (b) Intraband loss function calculated without local fields.

Figure 4

(a) $\mathrm{JDOS}(\mathbf{q},\omega )/q^2$ in ${\mathrm{Ca}}_2\mathrm{N}$ (solid lines) and in the HEG (dashed lines). The intensity has been normalized in order to be the same at the smallest $\mathbf{q}$. (b) Plasmon dispersion obtained from the JDOS in ${\mathrm{Ca}}_2\mathrm{N}$ (black line) and in the HEG (red line), and including matrix elements for ${\mathrm{Ca}}_2\mathrm{N}$ (blue line). The plasmon energy has been normalized to the energy of the smallest $\mathbf{q}$.

Figure 5

Loss function of ${\mathrm{Ca}}_2\mathrm{N}$ as a function of momentum transfer $\mathbf{q}$ ($x$ axis in the plot) along the $z$ direction normal to the planes up to third Brillouin zone calculated (a) with and (b) without local fields.

Figure 6

(a) Band structure of single-layer ${\mathrm{Ca}}_2\mathrm{N}$ in the 2D hexagonal Brillouin zone. The anionic bands are marked in red. (b) Crystal structure of a single-layer ${\mathrm{Ca}}_2\mathrm{N}$ (for the ball-and-stick representation, see Fig. 1). The charge associated to a wave function at the Fermi level is localized both inside the Ca-N-Ca layer and in two flat sheets parallel to the atomic plane.

Figure 7

Loss function of single-layer ${\mathrm{Ca}}_2\mathrm{N}$.

Figure 8

Plasmon dispersion in a ${\mathrm{Ca}}_2\mathrm{N}$ monolayer: full calculation, without interband contributions and without local-field effects.