Robust flat bands in $R{\mathrm{Co}}_5$ ($R=\mathrm{rare}$ earth) compounds

The mechanism to realize the peculiar flat bands generally existing in $R{\mathrm{Co}}_5$ ($R=\mathrm{rare}$ earth) compounds is clarified by analyzing the first-principles band structures and the tight-binding model. These flat bands are constructed from the localized eigenstates, the existence of which is guaranteed by the destructive interference of the intersite hopping among the $\mathrm{Co}\text{−}3d$ states at the kagome sites and those between the kagome and honeycomb sites. Their relative positions to other bands can be controlled by varying the lattice parameters keeping their dispersion almost flat, which suggests the possibility of flat-band engineering.

Figure 1

(a) Crystal structure of ${\mathrm{YCo}}_5$ and (b) its top view drawn using the vesta software [36]. Its first-principles band structures for the (c) majority and (d) minority spins. (e) Spin-unpolarized band structures obtained from the first-principles (solid black line) and the tight-binding-model (broken red line) calculations where the Fermi level of the latter is adjusted to have the flat band along the $\mathrm{\Gamma }-\mathrm{A}$ line placed in the same position as that in the former.

Figure 2

(a) Schematic picture of the localized eigenstate for the $k_x=k_y=0$ flat band. (b) Reflection and $C_3$ rotation, which generate the $C_{3\mathrm{v}}$ group. (c) Two reflections on each kagome site.

Figure 3

(a) Schematic picture of the localized eigenstate for the $k_z=0$ flat band. The signs of all the orbitals at the middle between the kagome and the honeycomb planes are shown. The sizes of the orbitals are depicted just schematically and the same sizes only mean similar weights. Shaded background is a guide to the eye. Panels (b) and (c) show two characteristic interference processes to realize the localization where a different color of an arrow means a different sign of the hopping amplitude. (d) Linear combination of the localized states in the honeycomb lattice [41].

Figure 4

(a) Exponentially decaying eigenstate with the definition of the site indices and the hopping amplitudes $t_l \left(l=1,2,3,4\right)$. (b) Hopping amplitudes for Fig. 3.

Figure 5

The ${\mathrm{YCo}}_5$ band structures of the majority spin with lattice parameters (a) increased or (b) decreased by 10% isotropically, and those with only the $c$ axis (c) increased or (d) decreased by 10%.