Single-valley engineering in graphene superlattices

The two inequivalent valleys in graphene are protected against long-range scattering potentials due to their large separation in momentum space. In tailored $\sqrt{3}N\times \sqrt{3}N$ or $3N\times 3N$ graphene superlattices, these two valleys are folded into $\mathrm{\Gamma }$ and coupled by Bragg scattering from periodic adsorption. We find that, for top-site adsorption, strong intervalley coupling closes the bulk gap from inversion symmetry breaking and leads to a single-valley metallic phase with quadratic band crossover. The degeneracy at the crossing point is protected by $C_{3v}$ symmetry. In addition, the emergence of pseudo-Zeeman field and valley-orbit coupling are also proposed, which provide the possibility of tuning valley polarization coherently in analogy to real spin for spintronics. Such valley manipulation mechanisms can also find applications in honeycomb photonic crystals. We also study the strong geometry-dependent influence of hollow- and bridge-site adatoms in the intervalley coupling.

Figure 1

Schematic representation of intervalley coupling adatom superlattices and their respective Brillouin zones. (a) and (b) are, respectively, primitive and reciprocal lattices for the top adsorption in $\sqrt{3}\times \sqrt{3}$ graphene supercells. The red lines represent the Brillouin zone of pristine graphene.

Figure 2

Topological transition from a quantum valley-Hall insulator to a single-valley phase as a function of the parameters $u_1$. Here, we set $u_2$ to be fixed with $u_2<0$ and $u_3=0$. $-2{\mathrm{\Delta }}_2$ corresponds to the local band gap from the intervalley scattering. ${\mathrm{\Delta }}^{\prime }$ measures the bulk (local) band gap from the competition between intervalley coupling and sublattice potentials. The progressive decrease of $u_1$ leads to a complete closure of the quantum valley-Hall gap and then transitions to the single-valley phase by reversing ${\mathrm{\Delta }}^{\prime }$.

Figure 3

Upper panel: schematic representation of honeycomb photonic crystals with a $\sqrt{3}\times \sqrt{3}$ periodicity. ${\vec{\alpha }}_1$ and ${\vec{\alpha }}_2$ denote the primitive vectors. The distance between nearest columns is set to be $a$ and the slab width is $d=0.1a$. $r_i$ $\left(i=1\text{–}3\right)$ label the radii of different columns. In our simulation, each column is chosen to be infinitely long. Lower panel: photonic band structures of transverse-magnetic modes along high-symmetry lines for different radii $r_1=0.18a$ (a), $0.23a$ (b), $0.28a$ (c), and $0.32a$ (d), respectively. Here, we set $r_2=0.25a$ and $r_3=0.18a$.

Figure 4

Primitive cells for hollow-site (a) and bridge-site (c) adsorptions. $\overrightarrow{{\alpha }_i}$ indicates the primitive lattice vectors for $1\times 1$ (in red) and $3\times 3$ (in black) graphene supercell. $u_1$ denotes the site energy induced by the adatoms. (b) Reciprocal lattices for $1\times 1$ (in red) and $3\times 3$ (in black) graphene supercell. (d), (e) Low-energy band structures for hollow- and bridge-site adsorptions. $\mathrm{\Delta }$ indicates the local gap from pseudo-Zeeman field in bridge adsorption.

Figure 5

Left panel: schematic of honeycomb-structured photonic crystals. The distance between nearest columns is set to be $a$ and the slab width is $d=0.1a$. $r_i$ $\left(i=1\text{–}3\right)$ label the radii of different columns. Right panel: the photonic band structure with uniform column radii and the slab widths where Dirac cones are present around the $K$ point and $K^{\prime }$ point and the later one is not shown for clarity of the figure.