Probing a divergent van Hove singularity of graphene with a ${\mathrm{Ca}}_2\mathrm{N}$ support: A layered electride as a solid-state dopant

Layered electrides, as typified by ${\mathrm{Ca}}_2\mathrm{N}$, are unconventional quasi-two-dimensional metals with low work functions, in which the conduction electrons are localized between the cation layers as well as outside the surface. We have investigated the electronic structure of the interface between a layered electride and another material, using graphene on ${\mathrm{Ca}}_2\mathrm{N}$ as an example. Our first-principles calculation shows that a graphene layer on ${\mathrm{Ca}}_2\mathrm{N}$ remains flat and is doped to an extremely high density of $n=5\times {10}^{14}{\mathrm{cm}}^{-2}$ with its Fermi level ($E_F$) aligned with the logarithmically divergent van Hove singularity (VHS) at a saddle point of graphene's ${\pi }^{*}$ band. This finding shows that graphene/${\mathrm{Ca}}_2\mathrm{N}$ is an ideal testing ground for the exploration of the many-body ground states, most notably superconducting states ($p+ip$ wave, $d$ wave, and $f$ wave), predicted to appear when $E_F$ is close to a VHS. The work function decreases abruptly upon monolayer attachment but reverts to that of ${\mathrm{Ca}}_2\mathrm{N}$ upon bilayer attachment. This peculiar behavior is explained in terms of the distinctive electronic structures of the constituent materials and their bonding.

Figure 1

(a) ${\mathrm{Ca}}_2\mathrm{N}$(001) slab, consisting of six LUs (18 layers), used in our study. It was optimized by relaxing the upper three LUs while keeping the lower three LUs frozen in the optimized structure for the bulk. (b) Band structure of the slab in (a) obtained by DFT along the symmetry lines of the 2D hexagonal Brillouin zone. The bands spanning the energy range between $-1.4$ and 1 eV are the anionic bands (indicated by the asterisk), which are either bulk states, having charge density mainly confined to the gaps between the LUs, or extra-surface states localized immediately outside the surfaces. The extra-surface states are highlighted by bold dots. The light blue shaded areas near the Fermi level indicate the continua of bulk energy bands projected onto the 2D surface Brillouin zone.

Figure 2

Stacking configurations considered in our calculation. The shaded areas indicate the unit cell and the spheres denote Ca ions in the top layer of the ${\mathrm{Ca}}_2\mathrm{N}$ slab. Configurations A and V: MLG/${\mathrm{Ca}}_2\mathrm{N}$(001) with the lines indicating the hexagonal lattice of graphene. Configurations Aa, Ab, and Va: BLG/${\mathrm{Ca}}_2\mathrm{N}$(001) with the solid (dashed) lines denoting the lattice of the inner (outer) graphene layer. Bernal stacking between the two graphene layers is assumed, with Aa and Ab derived from A, and Va from V. The red dashed lines at the left of A and Aa indicate the slice planes used to draw Fig. 4 and Figs. 7 and 7.

Figure 3

Layer-averaged electron densities for the structures investigated. (a) Total electron density for the ${\mathrm{Ca}}_2\mathrm{N}$ slab. The peaks indicate the positions of the ionic layers. (b) Partial electron density for the ${\mathrm{Ca}}_2\mathrm{N}$ slab. Note that the dominant peaks are located in the inter-LU gaps and also outside the surface ionic layer (extra-surface states indicated as ESS). (c) Partial electron density for MLG/${\mathrm{Ca}}_2\mathrm{N}$, where G1 indicates the graphene layer. (The G1 peak has two subpeaks corresponding to the two lobes of a carbon $2p_z$ orbital.) (d) Partial electron density for BLG/${\mathrm{Ca}}_2\mathrm{N}$, where G1 and G2 indicate the inner and outer graphene layers, respectively. (b)–(d) Calculated for electron energies between $-0.5$ and 0 eV.

Figure 4

(100) contour maps of the partial charge densities (electron energy between $-0.5$ and 0 eV) for (a) ${\mathrm{Ca}}_2\mathrm{N}$, (b) MLG/${\mathrm{Ca}}_2\mathrm{N}$, and (c) BLG/${\mathrm{Ca}}_2\mathrm{N}$. The slice plane is that indicated by red dashed lines at the left of A and Aa in Fig. 2. The color gradient is RGB with an electron density ranging from 0 (blue) to $3\times {10}^{-3}{\mathrm{bohr}}^{-3}$ (red). The densities on the contour lines are in a ratio of 3 starting from a minimum of ${10}^{-4}{\mathrm{bohr}}^{-3}$.

Figure 5

(a) Calculated band structure of MLG/${\mathrm{Ca}}_2\mathrm{N}$. The $\pi$ bands of graphene are highlighted in red with the radius of each dot proportional to the density of electrons for a given $k$ in the graphene layer. The light blue shaded areas near the Fermi level indicate the bulk energy bands projected onto the 2D surface Brillouin zone (continuous spectrum at each 2D $k$ point). (b) Partial density of states (PDOS) for graphene calculated from the band structure in (a). (c) Graphene bands in MLG/${\mathrm{Ca}}_2\mathrm{N}$ [red dots in (a)] compared with the bands calculated for an isolated MLG (black dots). The latter were calculated after removing the Ca and N atoms from the optimized structure of (a) while keeping the positions of the C atoms fixed. The obtained energy bands were downshifted so that the Dirac cone apex at $\mathrm{\Gamma }$ occurs at the same energy as in (a).

Figure 6

(a) Calculated band structure of BLG/${\mathrm{Ca}}_2\mathrm{N}$. The $\pi$ bands of the outer graphene layer are highlighted in blue while those of the inner graphene layer (in immediate contact with ${\mathrm{Ca}}_2\mathrm{N}$) are shown in red. The radius of each colored dot is proportional to the density of electrons at each $k$ in the corresponding graphene layer. The light blue shaded areas near the Fermi level indicate the bulk energy bands of ${\mathrm{Ca}}_2\mathrm{N}$ projected onto the 2D surface Brillouin zone. (b) and (c) Partial density of states (PDOS) for (b) the inner graphene layer and (c) the outer graphene layer calculated from the band structure of (a). (d) Graphene bands in BLG/${\mathrm{Ca}}_2\mathrm{N}$ [red and blue dots in (a)] compared with the bands calculated for an isolated bilayer graphene (black dots). The latter were obtained after removing the Ca and N atoms from the optimized structure while keeping the positions of the C atoms fixed. The obtained energies were downshifted to facilitate comparison with (a).

Figure 7

(a) (100) contour map of the difference electron density for MLG/${\mathrm{Ca}}_2\mathrm{N}$ defined as the difference between the electron density of the MLG/${\mathrm{Ca}}_2\mathrm{N}$ slab and the sum of the electron densities of the graphene layer and the ${\mathrm{Ca}}_2\mathrm{N}$ slab. D in the figure indicates where the depletion of electrons dominantly takes place. The gray scale ranges from $-5\times {10}^{-3}$ (black) to $3\times {10}^{-3}{\mathrm{bohr}}^{-3}$ (white), and the contours are in steps of $2.5\times {10}^{-3}{\mathrm{bohr}}^{-3}$ starting from a minimum of $-5\times {10}^{-3}$. (b) (100) contour map of the difference electron density for BLG/${\mathrm{Ca}}_2\mathrm{N}$ defined as the difference between the electron density of the BLG/${\mathrm{Ca}}_2\mathrm{N}$ slab and the sum of the electron densities of the graphene layer G2 and the MLG/${\mathrm{Ca}}_2\mathrm{N}$ slab. The gray scale ranges from $-5\times {10}^{-4}$ (black) to $3\times {10}^{-4}{\mathrm{bohr}}^{-3}$ (white), and the contours are in steps of $2.5\times {10}^{-4}{\mathrm{bohr}}^{-3}$ starting from a minimum of $-5\times {10}^{-4}{\mathrm{bohr}}^{-3}$. The arrows in (a) and (b) indicate the direction of electron transfer as a graphene layer is deposited. (c) Layer-averaged electrostatic potential $V\left(z\right)$ for MLG/${\mathrm{Ca}}_2\mathrm{N}$ minus the sum of $V\left(z\right)$ for graphene and ${\mathrm{Ca}}_2\mathrm{N}$. (d) Layer-averaged electrostatic potential $V\left(z\right)$ for BLG/${\mathrm{Ca}}_2\mathrm{N}$ minus the sum of $V\left(z\right)$ for graphene and MLG/${\mathrm{Ca}}_2\mathrm{N}$. In (c) and (d), $\mathrm{\Delta }\mathrm{\Phi }$ indicates the change in work function due to the attachment of the (last) graphene layer. The positions of the atomic layers are indicated by dashed vertical lines with G1 and G2 denoting the inner and outer graphene layers, respectively.

Figure 8

Energy band structures near $E_F$ for MLG/${\mathrm{Ca}}_2\mathrm{N}$ with stacking configurations (a) A and (b) V.

Figure 9

Energy band structures near $E_F$ for BLG/${\mathrm{Ca}}_2\mathrm{N}$ with stacking configurations (a) Aa, (b) Ab, and (c) Va.