Valley-Polarized Quantum Anomalous Hall Effect in Ferrimagnetic Honeycomb Lattices

The valley-polarized quantum anomalous Hall effect (VP-QAHE), which combines valleytronics and topology in one material, is of significant fundamental and practical importance in condensed-matter physics and materials science. In previous model studies, VP-QAHE occurs under strong Rashba spin-orbit coupling (SOC), which is an extrinsic effect. Here, using a low energy $k·p$ model, we propose a different mechanism of VP-QAHE by introducing an intrinsic staggered magnetic exchange field and develop a general picture of valley dependent band inversion in honeycomb lattice. Using first-principles calculation, this new mechanism is further demonstrated in the Co decorated In-triangle adlayer on a Si(111) surface. This system is equivalent to a ferrimagnetic honeycomb lattice, and the supported adlayer is experimentally more feasible in synthesis, thus exhibiting advantages over the existing studies based on Rashba SOC and free-standing sheets. The underlying physical mechanism is generally applicable, opening a new avenue for exploration of substrate supported VP-QAHE.

Figure 1

(a) Ferrimagnetic honeycomb lattice (blue: sublattice $A$, spin-up; red: sublattice $B$, spin-down). (b) Schematic plot of low energy band at two valleys before (left panel) and after (right panel) including atomic SOC. The index in the left panel is ($⟨{\sigma }_z⟩$, $⟨s_z⟩$) of each band at the valley. (c) Variation of band gap at the two valleys $E_g/{\lambda }_{\mathrm{SO}}$ with respect to $m_B/{\lambda }_{\mathrm{SO}}$. (d) Topological phase diagram of $m_B$ and ${\lambda }_{\mathrm{SO}}$ (the dashed line is ${\lambda }_{\mathrm{SO}}=|m_B|$).

Figure 2

(a) The atomic structure of the Co decorated In-triangle adlayer on Si. (b) Isosurface ($0.005|e|/{\AA }^3$) of spin density with blue and green color representing the up and down spins, respectively. (c) The first Brillouin zone.

Figure 3

(a) Band structure (with SOC) with color of each state representing out-of-plane $⟨s_z⟩$. (b) and (c) Enlargements of band structure near the Fermi level along the $\mathrm{\Gamma }\to K$ and $K\to K^{\prime }$ paths, respectively. In (b) $\mathrm{\Delta }$ denotes the point with the narrowest band gap.

Figure 4

(a) Berry curvature along the high symmetry $k$ path. (Inset) Schematic plot of VP-QAHE with their edge conductance. (b) $k$-resolved Berry curvature. (c) Anomalous Hall conductance variation of chemical potential.