First-principles calculation of the graphene Dirac band on semi-infinite Ir(111)

We study the energy dispersion relation of the $\pi$ and ${\pi }^{*}$ bands in epitaxial monolayer graphene on a semi-infinite Ir(111) substrate by a first-principles density-functional calculation. For this purpose, we employ a realistic surface structure in which the $(10\times 10)$ unit cell of graphene matches a $(9\times 9)$ cell of Ir(111). We determine the surface geometry by using a slab model containing four Ir layers, and the optimized structure is used as input for the subsequent surface embedded Green's function calculation. By taking advantage of semi-infinite calculations, we discuss mini energy gaps at the crossing of the $\pi$ band and its replicas, the Rashba-type spin splitting of the $\pi$ and ${\pi }^{*}$ bands, and also the energy width of both bands arising from interactions with the energy continuum of bulk Ir bands.

Figure 1

(a) Top and (b) side views of the $(10\times 10)$ graphene monolayer on a $(9\times 9)$ Ir(111) slab containing four Ir layers. The red parallelogram indicates a $(9\times 9)$ supercell of the substrate. Three dashed circles encompass regions where carbon hexagons are located at atop, hcp, and fcc adsorption sites.

Figure 2

Intensity plots of (a) $\rho (\mathbf{k},\varepsilon )$, (b) $s_y(\mathbf{k},\varepsilon )$, and (c) $s_z(\mathbf{k},\varepsilon )$ calculated in a first-layer Ir muffin-tin sphere with radius $R=2.48$ bohrs for semi-infinite Ir(111) along the $k_x$ direction. The $K$ point of Ir(111) is located at ${\mathbf{k}}_s=(k_s,0)$. Color varies linearly with ${log}_{10}\left|\rho (\mathbf{k},\epsilon )\right|$ in panel (a) and linearly with $|s_{\alpha }(\mathbf{k},\epsilon )|$ in panels (b) and (c) (see color scale bars). The imaginary energy $\gamma =2\mathrm{meV}$. The inset in (c) illustrates the surface Brillouin zones of pristine Ir(111) (dashed hexagon) and free-standing graphene (solid hexagon).

Figure 3

(a), (d), (g) $\rho (\mathbf{k},\varepsilon )$ averaged over two inequivalent C atoms. (b), (e), (h) $\rho (\mathbf{k},\varepsilon )$ for a first-layer Ir atom. (c), (f), (i) $s_z(\mathbf{k},\varepsilon )$ for a first-layer Ir atom. Panels (a)–(c), (d)–(f), and (g)–(i) correspond to the atop, hcp, and fcc adsorption geometries, respectively. The same color scales as those in Fig. 2 are used. Muffin-tin sphere radii are $R=1.33$ and 2.48 bohrs for C and Ir atoms, respectively. The imaginary energy $\gamma =2\mathrm{meV}$. The inset in (b) displays a magnified view of the DOS at the band crossing. The inset in (c) displays $s_z(\mathbf{k},\varepsilon )$ in a small energy interval containing two peaks of the spin-split $\pi$ band marked by a red arrow.

Figure 4

Intensity plot of $\tilde{\rho }(\mathbf{k},\varepsilon )$ for the $(10\times 10)$ graphene overlayer on Ir(111). Color varies linearly with $log|\tilde{\rho }(\mathbf{k},\varepsilon )|$ (see color scale bar). (a) $\mathbf{k}=(k_x,0)$ with $0.75k_D\le k_x\le 1.25k_D$ where $k_x=k_s(=0.9k_D)$ and $k_x=k_D$ correspond to the $K$ points of Ir(111) and graphene, respectively. (b) $\mathbf{k}=(k_D,k_y)$ with $0\le k_y\le 0.1k_D$. The muffin-tin sphere radius of C atoms is chosen as $R=1.33$ bohrs. $\mathrm{\Delta }\varepsilon =20\mathrm{meV}$ and $\gamma =2\mathrm{meV}$.

Figure 5

(a) $\tilde{\rho }(\mathbf{k},\varepsilon )$ of the $(10\times 10)$ graphene overlayer on Ir(111) at 11 $\mathbf{k}$ points on the $k_x$ axis. (b) ${\tilde{s}}_y(\mathbf{k},\varepsilon )$ (dotted lines) and ${\tilde{s}}_z(\mathbf{k},\varepsilon )$ (solid lines) at the same $k_x$'s as in (a). $\mathrm{\Delta }\varepsilon =20\mathrm{meV}$ and $\gamma =2\mathrm{meV}$ in both panels. The small number below each curve indicates $k_x/k_D$ where ${\mathbf{k}}_D=(k_D,0)$ is the $K$ point of graphene. ${\tilde{s}}_y(\mathbf{k},\varepsilon )$ and ${\tilde{s}}_z(\mathbf{k},\varepsilon )$ at a given $k_x$ are vertically shifted for better visibility. The insets (i) and (ii) in panel (a) show the $\pi$ and ${\pi }^{*}$ band peaks at $k_x=0.925k_D$ calculated with $\mathrm{\Delta }\varepsilon =2\mathrm{meV}$ and $\gamma =1\mathrm{meV}$, respectively.

Figure 6

Charge density in the energy range 0.32 eV $\le \varepsilon -E_F\le$ 0.54 eV at the $K$ point of graphene corresponding to the FWHM of the broad DOS peak at $E_D$ on a vertical cut plane along the diagonal line $AB$ in Fig. 1.

Figure 7

Red line indicates a small portion of the first BZ of the $(1\times 1)$ graphene near ${\mathbf{k}}_D=(k_D,0)$, the $K$ point on the $k_x$ axis. Small hexagons are mini BZs of the $(10\times 10)$ supercell whose centers are chosen to be at ${\mathbf{k}}_D+{\mathbf{G}}_i$ where ${\mathbf{G}}_i$ is a reciprocal lattice vector of the supercell. Replica Dirac cones centered at neighboring ${\mathbf{k}}_D+{\mathbf{G}}_i$'s interact on the boundaries of the mini BZs, leading to the opening of a mini energy gap. Small numbers along the $k_x$ axis indicate $k_x/k_D$.

Figure 8

$\tilde{\rho }(\mathbf{k},\varepsilon )$ at (a) $\mathbf{k}={\mathbf{k}}_s$, (c) $\mathbf{k}={\mathbf{k}}_1$, and (e) $\mathbf{k}={\mathbf{k}}_2$. ${\tilde{s}}_z(\mathbf{k},\varepsilon )$ at (b) $\mathbf{k}={\mathbf{k}}_s$, (d) $\mathbf{k}={\mathbf{k}}_1$, and (f) $\mathbf{k}={\mathbf{k}}_2$. In all panels, $\mathrm{\Delta }\varepsilon =2\mathrm{meV}$ and $\gamma =1\mathrm{meV}$.