Completely flat bands and fully localized states on surfaces of anisotropic diamond-lattice models

We discuss flat-band surface states on the (111) surface in the tight-binding model with nearest-neighbor hopping on the diamond lattice, in analogy to the flat-band edge states in graphene with a zigzag edge. The bulk band is gapless, and the gap closes along a loop in the Brillouin zone. The verge of the flat-band surface states is identical with this gap-closing loop projected onto the surface plane. When anisotropies in the hopping integrals increase, the bulk gap-closing points move and the distribution of the flat-band states expands in the Brillouin zone. Then when the anisotropy is sufficiently large, the surface flat bands cover the whole Brillouin zone. Because of the completely flat bands, we can construct surface-state wave functions, which are localized in all the three directions.

Figure 1

(a) Schematic of the honeycomb-lattice structure. The dotted line shows a choice of unit cell with translational symmetry along the edge. The arrows show the directions along zigzag, armchair, and Klein edges. (b) shows the first BZ of the honeycomb lattice. Dots at the zone corners show the gap-closing points for the graphene model ($t_1=t_2=t_3=1$). Namely, the bulk bands become gapless at $K$ and $K^{\prime }$ points, when the hopping integral is isotropic. By increasing $t_1$ from unity, the gap-closing points move away from $K$ and $K^{\prime }$ points as shown by the arrows. The flat-band edge states expand in the BZ as the bulk gap-closing points move. (c) shows the dispersions for ribbon geometry for $t_1=1$ and $2.2$ at $t_2=t_3=1$. The dispersions are shown for zigzag, armchair and Klein edges. In zigzag and Klein edges, flat bands appear at the zero energy.

Figure 2

(a) Schematic of a diamond lattice. $A$ (gray) and $B$ (black) atoms denote the two sublattices. $\tau$s represent vectors connecting nearest-neighbor atoms. (b) The BZ of the diamond lattice structure within $k_{x,y,z}>0$.

Figure 3

Band structure for the diamond lattice model with the $\left(111\right)$ surface, for the two cases: (a) $t_1=t$, $t_2=t_3=t_4=1$ and (b) $t_3=t$, $t_1=t_2=t_4=1$. (a1) and (b1) show the respective band structures close to zero energy for (a) and (b). The thick lines represent the first BZ. (a2) and (b2) show the region with the surface flat bands at zero energy. In (c), the FS at zero energy in the 3D bulk BZ projected on the $\left(111\right)$ plane for the cases (a) and (b).

Figure 4

The FSs at zero energy in the 3D bulk BZ projected on the $\left(111\right)$ plane for $t_2=t_3=t_4=1$ and $t_1=t=0.6,1,1.4,2.2,3.0$ are shown. For $t>1$, the loop shrinks with increase in $t$, and at $t=3.0$ the loop becomes a dot.

Figure 5

Dispersion near zero energy at $t=3.4$ for the slab geometry. The number $N$ of unit cells to the direction normal to the surface is $N=20$ in (a) and 40 in (b).

Figure 6

The FSs in the 3D bulk BZ for $t_2=t_3=t_4=1$ and $t_1=t=0.8$, 1, 1.2 are shown. At $t=0.8$ (blue) the FSs consist of two open orbits. On the other hand at $t=1.2$ (red), the FS becomes one closed orbit in the BZ. To illustrate the topological transition of the FS at $t=1$ (green), the FSs around one of the $X$ points are magnified in the inset in the extended zone scheme.

Figure 7

Trajectories of $(h_x,h_y)$ by varying $k_x^{\prime }$ in (a) and (b) or $k_3$ in (c) for the honeycomb-lattice model with zigzag (a), with Klein (b) edges, and in the diamond-lattice model with the $\left(111\right)$ surface (c). In these trajectories, $k_y$, $k_1$, and $k_2$ are fixed.

Figure 8

Schematic of spatial distribution of the flat-band surface states. The circles are the $A$ sublattice sites, while the amplitudes on $B$ sublattices are zero and are omitted. The dotted circles show that their amplitudes of the wave function are zero. The line thickness of the circle shows the magnitude of the amplitude. The top atom has the largest amplitude, and the distribution of the wave function spatially spreads toward the interior with exponential decay. This picture of the fully localized states applies both to the honeycomb lattice with the zigzag edge and to the diamond lattice with the (111) surface.

Figure 9

Schematic of the grids for the shortest path problem. The mover is at first at the origin $\mathcal{O}$, and it moves along ${\mathbf{e}}_i$ with the probability $P_i$, where ${\left({\mathbf{e}}_i\right)}_j={\delta }_{ij}$.