Surface electron states on the quasi-two-dimensional excess-electron compounds ${\mathrm{Ca}}_2\mathrm{N}$ and ${\mathrm{Y}}_2\mathrm{C}$

Compounds having excess electrons from the formal valence viewpoint (electrides) are a new class of materials, which often take low-dimensional structures. We studied the (001) surface electronic structures of quasi-two-dimensional electrides ${\mathrm{Ca}}_2\mathrm{N}$ and ${\mathrm{Y}}_2\mathrm{C}$ by density functional theory using a slab model. Both materials were found to have a clean surface state well separated in energy from the bulk states. Furthermore, this state virtually floats above the surface and may be considered to be a hallmark of two-dimensional electrides. For ${\mathrm{Ca}}_2\mathrm{N}$, a tight-binding model in the Wannier representation was derived and analyzed, from which we concluded that the surface state, described by extra-surface $s$-like orbitals, is a Tamm state originating from an abrupt increase in potential energy at the surface.

Figure 1

Left panel: Crystal structure of ${\mathrm{Ca}}_2\mathrm{N}$ and ${\mathrm{Y}}_2\mathrm{C}$ (rhombohedral, $R\overline{3}m$), with the cations (metal) and anions shown by large orange spheres and small blue spheres, respectively. A conventional unit cell contains three layer units (LUs) arranged in ABC stacking [LU(I), LU(II), and LU(III)]. For ease of display, a supercell comprising $3\times 3\times 3$ conventional cells is drawn. The slab used for our surface calculation is a $1\times 1\times 3$ supercell, i.e., its height is the same as in this illustration but the lateral size is scaled by a factor of $\frac{1}{3}$ in the $a$ and $b$ directions. Right panel: Bulk and (001) surface Brillouin zones.

Figure 2

Electronic band structure (Kohn-Sham energies plotted against 2D wave number $\mathit{k}$) of bulk (a) ${\mathrm{Ca}}_2\mathrm{N}$ and (b) ${\mathrm{Y}}_2\mathrm{C}$ calculated by DFT projected onto the (001) surface Brillouin zone (right panel in Fig. 1).

Figure 3

Electronic band structures calculated by DFT for the nine-LU (001) slabs of (a) ${\mathrm{Ca}}_2\mathrm{N}$ and (b) ${\mathrm{Y}}_2\mathrm{C}$ (black points). Superimposed on these points are (i) blue dots whose radii indicate the probability of the electron being in the LU(I)-LU(II) gap and (ii) red dots, the radii of which are proportional to the probability of the electron being immediately outside the surface. At lower energies, the surface bands derived from nitrogen $p$ states are highlighted in green [32].

Figure 4

(a) and (c) Layer-averaged PED for the energy range $-0.1$ to 0 eV (red solid line) and total VED (black dashed line) calculated for the ${\mathrm{Ca}}_2\mathrm{N}$ and ${\mathrm{Y}}_2\mathrm{C}$ slabs, respectively. The peaks arising from the extra-surface states are indicated by asterisks. In (a), the positions of the centers of the Wannier orbitals $|s0\rangle$ and $|s1\rangle$ are indicated in the upper right. (b) and (d) Layer-averaged PEDs at $k$ points A (red solid line) and B (black dashed line) in Fig. 3 for ${\mathrm{Ca}}_2\mathrm{N}$ and ${\mathrm{Y}}_2\mathrm{C}$ slabs, respectively.

Figure 5

Electron density contours in the [100] plane for (001) slabs of ${\mathrm{Ca}}_2\mathrm{N}$ and ${\mathrm{Y}}_2\mathrm{C}$. (a) VED and (b) PED for ${\mathrm{Ca}}_2\mathrm{N}$. (c) VED and (d) PED for ${\mathrm{Y}}_2\mathrm{C}$. The PEDs are calculated for the energy range $-0.1\text{eV}<E<0$ eV. The symbols $M$ and $X$ denote a metal ion (Ca/Y) and an anion (N/C), respectively. The crosses at the top of (b) indicate the positions of the Wannier orbital $|s0\rangle$ obtained in Sec. 4. The densities on successive contours are in the ratio of $\sqrt{3}$.

Figure 6

Distances between near-surface metal layers in the relaxed slab. Their changes relative to the bulk values are shown in parentheses. The inter-LU probability, as indicated by the blue dots in Fig. 2, is calculated by integrating the square of the wave function ${\mathrm{\Psi }}_{nk}^2$ in a sphere of 1.3 Å radius at the interstitial position A. This position corresponds to the fractional coordinates $\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ in the primitive unit cell $\left[\right(0,0,0.5)$ in the conventional unit cell] in the bulk with Wyckoff position $b$ and site symmetry $\overline{3}m$. Similarly, the extra-surface probability indicated by the red dots in Fig. 3 is obtained by placing a sphere of the same radius at an “interstitial” position B of a hypothetical infinite crystal, which is equivalent to position ${\mathrm{B}}^{\prime }$.

Figure 7

Calculated band structure for a single LU of (a) ${\mathrm{Ca}}_2\mathrm{N}$ and (b) ${\mathrm{Y}}_2\mathrm{C}$. The layer-averaged PED for energy $-0.1\text{eV}<E<0$ eV (red solid line) and the total VED (black dotted line) are shown in the insets.

Figure 8

(a) Energy band structure of the ${\mathrm{Ca}}_2\mathrm{N}$ (001) slab. The black dots denote the result of a DFT calculation [same as Fig. 3]. The red dots show the energy bands obtained by diagonalizing the tight-binding Hamiltonian with maximally localized WFs as the basis. (b) Positions of Wannier centers indicated by crosses. The large (orange) spheres and small (blue) spheres denote Ca and N atoms, respectively.

Figure 9

Isosurface graphics of obtained maximally localized WFs. Only the functions located near the surface are shown. $|p_{x}1\rangle$, $|p_{y}1\rangle$, $|p_{z}1\rangle$: $p$-like WFs centered at the nitrogen site in LU(I). $|s0\rangle$: $s$-like WF outside the surface, $|s1\rangle$: $s$-like WF slightly off the interstitial site in the LU(I)-LU(II) gap.

Figure 10

On-site energies $E_{\gamma Z}$ of the Wannier orbitals (diagonal elements of the tight-binding Hamiltonian) plotted against the $z$ coordinates of their centers. The $s$ orbitals, $p_{x,y}$ orbitals, and $p_z$ orbitals are indicated by red circles, black crosses, and black triangles, respectively.

Figure 11

Weight $C_{nk,\gamma Z}^2$ ($\gamma =s,p_x,p_y,p_z$) of WFs in the Bloch function expansion [Eq. (3)] at various $(k,n)$ plotted against the Wannier center coordinate $Z$. The $s$ orbitals, $p_{x,y}$ orbitals, and $p_z$ orbitals are indicated by red circles, black crosses, and black triangles, respectively. (a)–(d) correspond to $(k,n)$ at points A–D, respectively, in Fig. 8.