Hidden kagome-lattice picture and origin of high conductivity in delafossite ${\mathrm{PtCoO}}_2$

We study the electronic structure of delafossite ${\mathrm{PtCoO}}_2$ to elucidate its extremely small resistivity and high mobility. The band exhibits steep dispersion near the Fermi level despite the fact that it is formed mainly by Pt $d$ orbitals that are typically localized. We propose a picture based on two hidden kagome-lattice-like electronic structures: one originating from Pt $s+p_x/p_y$ orbitals, and the other from Pt $d_{3z^2-r^2}+d_{xy}/d_{x^2-y^2}$ orbitals, each placed on the bonds of the triangular lattice. In particular, we find that the underlying Pt $s+p_x/p_y$ bands actually determine the steepness of the original dispersion, so that the large Fermi velocity can be attributed to the large width of the Pt $s+p_x/p_y$ band. In addition, the kagome-like electronic structure gives rise to “orbital-momentum locking” on the Fermi surface, which reduces the electron scattering by impurities. We conclude that the combination of the large Fermi velocity and the orbital-momentum locking is likely to be the origin of the extremely small resistivity in ${\mathrm{PtCoO}}_2$.

Figure 1

Schematic figure of the orbital-momentum locking. The figure shows how the orbital character varies along the Fermi surface.

Figure 2

The crystal structure of ${\mathrm{PtCoO}}_2$ depicted using vesta [15]. The inset shows the triangular lattice of Pt atoms.

Figure 3

(a) First-principles band structure of ${\mathrm{PtCoO}}_2$. The thickness represents the strength of the $d$-orbital character. (b) The band structure of the 20-orbital tight-binding model. (c) The density of states of ${\mathrm{PtCoO}}_2$.

Figure 4

Bands that consist solely of (a) Pt $s$, (b) Pt $p_x/p_y$, (c) Pt $d_{xy}/d_{x^2-y^2}$, or (d) Pt $d_{3z^2-r^2}$ orbital components, extracted from the 20-orbital model derived from the first-principles band structure.

Figure 5

Band structure of models that consist of (a) Pt $s$, $p_x/p_y$, $d_{3z^2-r^2}$, $d_{xy}/d_{x^2-y^2}$, and $\mathrm{O}{p}_z$; (b) Pt $s$ and $p_x/p_y$; and (c) Pt $d_{3z^2-r^2}$ and $d_{xy}/d_{x^2-y^2}$ orbitals. Dashed circles in (a) denote the steep band intersecting the Fermi level.

Figure 6

(a)–(c) Evolution of the $s+p_x/p_y$ band structure upon varying the on-site energy level of the $s$ orbital. The thickness represents the strength of the $s$-orbital component.

Figure 7

(a) First-principles band structure of triangular lattice Si. (b) $s+p_x/p_y$ three-orbital model derived from the first-principles band structure.

Figure 8

(a) Wannier orbitals of the $s+p_x/p_y$ three-orbital model of triangular lattice Si depicted using vesta [15]. (b) The center of the Wannier orbitals forms a kagome lattice.

Figure 9

The relation between the kagome lattice with nearest-neighbor hopping only and the Si $s+p_x/p_y$ model is shown by adding the distant hoppings one by one. The values of the hopping integrals are shown in Fig. 8.

Figure 10

The similarity between the band structure of Pt $s+p_x/p_y$ (right) and that of Pt $d_{3z^2-r^2}+d_{xy}/d_{x^2-y^2}$ (left) is shown.

Figure 11

Wave function of the original kagome lattice plotted on a real-space unit cell. The large circle is a certain energy contour that encircles the $\mathrm{\Gamma }$ point, and the wave function is plotted along this contour. The radius of the circle represents the weight on each site, and the color denotes the sign (red: negative, blue: positive).

Figure 12

Weight of each orbital along the Fermi surface of ${\mathrm{PtCoO}}_2$. The definition of the angle $\theta$ is given in the inset.

Figure 13

The band structure of the six-orbital model with the spin-orbit coupling included. The blue line represents the dispersion of the single-orbital model, which is obtained by extracting the band intersecting the Fermi level (see the text).

Figure 14

(a) The absolute value of the imaginary part of the self-energy for the six-orbital and the single-orbital models. (b) The inner product of the wave functions on the Fermi surface plotted as a function of the angle $\theta$ measured from the $\mathrm{\Gamma }-M$ direction taken as ${\theta }^{\prime }=0$.